# Minkowski norm and Hessian isometry induced by an isoparametric foliation on
the unit sphere

Ming Xu  Ming Xu
School of Mathematical Sciences, Capital Normal University, Beijing 100048, P.
R. China<EMAIL_ADDRESS>
###### Abstract.

Let $M_{t}$ be an isoparametric foliation on the unit sphere
$(S^{n-1}(1),g^{\mathrm{st}})$ with $d$ principal curvatures. Using the
spherical coordinates induced by $M_{t}$, we construct a Minkowski norm with
the presentation $F=r\sqrt{2f(t)}$, which generalizes the notions of
$(\alpha,\beta)$-norm and $(\alpha_{1},\alpha_{2})$-norm. Using the technique
of spherical local frame, we give an exact and explicit answer for the
question when $F=r\sqrt{2f(t)}$ really defines a Minkowski norm. Using the
similar technique, we study the Hessian isometry $\Phi$ between two Minkowski
norms induced by $M_{t}$, which preserves the orientation and fixes the
spherical $\xi$-coordinates. There are two ways to describe this $\Phi$,
either by a system of ODEs, or by its restriction to any normal plane for
$M_{t}$, which is then reduced to a Hessian isometry between Minkowski norms
on $\mathbb{R}^{2}$ satisfying certain symmetry and (d)-properties. When
$d>2$, we prove this $\Phi$ can be obtained by gluing positive scalar
multiplications and compositions between the Legendre transformation and
positive scalar multiplications, so it must satisfy the (d)-property for any
orthogonal decomposition
$\mathbb{R}^{n}=\mathbf{V}^{\prime}+\mathbf{V}^{\prime\prime}$, i.e., for any
nonzero $x=x^{\prime}+x^{\prime\prime}$ and
$\Phi(x)=\overline{x}=\overline{x}^{\prime}+\overline{x}^{\prime\prime}$, with
$x^{\prime},\overline{x}^{\prime}\in\mathbf{V}^{\prime}$ and
$x^{\prime\prime},\overline{x}^{\prime\prime}\in\mathbf{V}^{\prime\prime}$, we
have
$g_{x}^{F_{1}}(x^{\prime\prime},x)=g_{\overline{x}}^{F_{2}}(\overline{x}^{\prime\prime},\overline{x})$.
As byproducts, we prove the following results. On the indicatrix $(S_{F},g)$,
where $F$ is a Minkowski norm induced by $M_{t}$ and $g$ is the Hessian
metric, the foliation $N_{t}=S_{F}\cap\mathbb{R}_{>0}M_{0}$ is isoparametric.
Laugwitz Conjecture is valid for a Minkowski norm $F$ induced by $M_{t}$,
i.e., if its Hessian metric $g$ is flat on $\mathbb{R}^{n}\backslash\\{0\\}$
with $n>2$, then $F$ is Euclidean.

Mathematics Subject Classification (2010): 52A20, 53C21, 53C40

Key words: Minkowski norm, Hessian isometry, Hessian metric, isoparametric
foliation, Laugwitz Conjecture, Legendre transformation

##  1\. Introduction

The classification of isoparametric foliations on the unit sphere
$(S^{n-1}(1),g^{\mathrm{st}})$ (if not otherwise specified, we will always
assume $n>2$) has been one of the most important geometric problems [58], with
a history of eighty years since the time of E. Cartan [5, 6]. There were many
remarkable progress [8, 9, 12, 16, 24, 33, 46], and recently it was completely
solved by Q.S. Chi [10]. Meanwhile, researchers are eager to find applications
and generalizations of this theory in geometry and topology. For example, its
applications in Riemannian geometry and differential topology are concerned in
[23, 35, 38, 52, 53]. Its generalization, the equifocal hypersurface, is
studied in [22, 48, 51]. Its generalization to Finsler geometry is studied in
[25, 27, 28, 55, 57]. More references can be found in the survey papers [20,
37, 49, 54].

In this paper, we consider how to generalize and apply the isoparametric
foliation on the unit spheres to the Hessian geometry [43] for Minkowski
norms. This work is inspired by the recent cowork [56] with V. Matveev, which
implies the interesting connections to the study of Laugwitz conjecture [29]
in convex geometry [40] and Landsberg Unicorn Conjecture [32, 42] in Finsler
geometry [4].

In this paper, we only consider smooth and strongly convex Minkowski norms on
finite dimensional real vector spaces [4]. For example, a Minkowski norm on
$\mathbb{R}^{n}$ with $n\geq 2$ is a continuous function
$F:\mathbb{R}^{n}\rightarrow\mathbb{R}_{\geq 0}$ which is positive and smooth
on $\mathbb{R}^{n}\backslash\\{0\\}$, and satisfies the positive 1-homogeneity
and the strong convexity (see [4] or Section 3.1). Then the Hessian of
$E=\tfrac{1}{2}F^{2}$, is positive definite at each nonzero $x$, which defines
a Riemannian metric $g={\rm d}^{2}E$ on $\mathbb{R}^{n}\backslash\\{0\\}$. For
simplicity, we call it the Hessian metric of $F$.

Since the Minkowski norm $F$ is one-to-one determined by its indicatrix
$S_{F}=\\{x\in\mathbb{R}^{n}|F(x)=1\\}$. The geometric properties of the
Hessian metric $g$ or its restriction to $S_{F}$ help us understand the
convexity of the domain enclosed by $S_{F}$. See [40, 43] for more discussion
on the relation between Hessian geometry and Convex geometry.

Notice that this is only one important model in more general Hessian geometry.
Hessian geometers have many other sources for the function $E$ to construct
the metric [21, 29, 30, 44], toric Kähler geometry, infinite dimensional
integrable system of hydrodynamic type, affine geometry of hypersurfaces,
information geometry, etc.. More involved discussion for Hessian geometry can
be found in [43] and references therein.

Now we come back to an isoparametric foliation $M_{t}$ on the unit sphere
$(S^{n-1}(1),g^{\mathrm{st}})\subset\mathbb{R}^{n}$. Here we parametrize
$M_{t}$ such that
$t=\mathrm{dist}_{S^{n-1}(1)}(M_{t},M_{0})\in[0,\tfrac{\pi}{d}]$, where
$M_{0}$ and $M_{\pi/d}$ are the two focal submanifolds, and
$d\in\\{1,2,3,4,6\\}$ is the number of principal curvatures for each $M_{t}$
with $t\in(0,\tfrac{\pi}{d})$ in $(S^{n-1}(1),g^{\mathrm{st}})$ [34].
Associated with $M_{t}$, we can define the (generalized) spherical coordinates
$(r,t,\xi)\in\mathbb{R}_{>0}\times(0,\tfrac{\pi}{d})\times M_{\pi/2d}$, i.e.,
$x=(r,t,\xi)$ when $|x|=r$, $x/|x|\in M_{t}$ and there exists a normal
geodesic segment in $(S^{n-1}(1),g^{\mathrm{st}})$ for this foliation, which
connects $x/|x|$ to $\xi$ without passing the focal submanifolds. Further
more, we introduce spherical local frame induced by $M_{t}$ (see Section 2.5),
with which the standard flat metric $g^{\mathrm{st}}$ on
$\mathbb{R}^{n}\backslash\\{0\\}$ and its Levi-Civita connection can be
explicitly calculated.

We can use the foliation $M_{t}$ to define a Minkowski norm $F$ on
$\mathbb{R}^{n}$, such that the restriction of $F$ to each $M_{t}$ is a
constant function. We will simply call it a Minkowski norm induced by $M_{t}$.
When $d=1$ or $2$, the induced $F$ admits a linear $SO(n-1)$\- or $O(k)\times
O(n-k)$-invariancy, and is called an $(\alpha,\beta)$-norm or
$(\alpha_{1},\alpha_{2})$-norm in some literature [14, 15]. These norms have
attracted many attentions of Finsler geometers [26, 31]. However, the induced
Minkowski norms when $d>2$ have been rarely studied.

Using the spherical $r$\- and $t$-coordinates, the induced Minkowski norm $F$
can be presented as $F=r\sqrt{2f(t)}$. A natural and important question is the
following:

######  Question 1.1.

When does $F=r\sqrt{2f(t)}$ define a Minkowski norm induced by $M_{t}$?

Notice that, besides the issue of strong convexity, the smoothness of
$F=r\sqrt{2f(t)}$ at $\mathbb{R}_{>0}M_{0}$ and $\mathbb{R}_{>0}M_{\pi/d}$ is
also subtle and crucial. We use the spherical local frame to calculate the
Hessian of $E=\tfrac{1}{2}F^{2}=r^{2}f(t)$ as in [56], and then completely
answer Question 1.1 by the following theorem.

Theorem A. The spherical coordinates presentation $F=r\sqrt{2f(t)}$ defines a
Minkowski norm induced by $M_{t}$ if and only if $f(t)$ can be extended to a
positive smooth $D_{2d}$-invariant function on $\mathbb{R}/(2\mathbb{Z}\pi)$
which satisfies

$2f(t)\tfrac{{\rm d}^{2}}{{\rm d}t^{2}}f(t)-\left(\tfrac{{\rm d}}{{\rm
d}t}f(t)\right)^{2}+4f(t)^{2}>0$

everywhere, i.e., the polar coordinates presentation
$\overline{F}=r\sqrt{2f(t)}$ defines a $D_{2d}$-invariant Minkowski norm on
$\mathbb{R}^{2}$.

Here $\mathbb{R}^{2}$ can be identified with any normal plane $\mathbf{V}$ for
$M_{t}$ (i.e., $\mathbf{V}\cap S^{n-1}(1)$ is a normal geodesic for $M_{t}$),
and $D_{2d}$ is the group $\mathbb{Z}_{2}$ when $d=1$ and the dihedral group
when $d>1$, which can be interpreted as a Weyl group. See Section 2.4 for its
explicit description and its action on $\mathbb{R}^{2}$ or
$\mathbb{R}/(2\mathbb{Z}\pi)$.

Theorem A is a reformulation of Theorem 3.1. Its direct corollaries, Corollary
3.2 and Corollary 3.3, where we take $d=1$ and $2$, reprove some known results
for Minkowski norms of $(\alpha,\beta)$\- and $(\alpha_{1},\alpha_{2})$-types
[11, 15].

Let $F=r\sqrt{2f(t)}$ be a Minkowski norm induced by $M_{t}$. Then on its
indicatrix $S_{F}$, there is a foliation $N_{t}=S_{F}\cap\mathbb{R}_{>0}M_{t}$
induced by $M_{t}$. Using the technique of spherical local frame again, we
prove the following theorem (see Theorem 3.7).

Theorem B. Let $F$ be a Minkowski norm induced by the isoparametric foliation
$M_{t}$ on $(S^{n-1}(1),g^{\mathrm{st}})$ and $g$ its Hessian metric. Then the
foliation $N_{t}=S_{F}\cap\mathbb{R}_{>0}M_{t}$ on $(S_{F},g)$ is
isoparametric.

Theorem B provides more examples of isoparametric foliations. Indeed, when
$M_{t}$ is homogeneous, i.e., it is induced by the isometric cohomogeneity one
action of some compact connected Lie group $G$ (see [24, 45, 50] for its
classification), the isometric $G$-action on $(S_{F},g)$ is also of
cohomogeneity one. So the $G$-orbits $N_{t}$ provide an isoparametric
foliation on $(S_{F},g)$. Though this shortcut to Theorem B is not valid for
inhomogeneous $M_{t}$ of OT-FKM type [18, 36], it provides the most crucial
hint, and it inspires us to more generally study the Hessian isometries
between Minkowski norms. It is also remarkable that similar correspondence has
been found for isoparametric foliations on smooth homotopy spheres (see
Theorem 1.1 in [19]), where topology rather than geometry or Lie theory plays
the main role.

Let $F_{1}$ and $F_{2}$ be two Minkowski norms on $\mathbb{R}^{n}$ with $n\geq
2$, and $g_{1}$ and $g_{2}$ their Hessian metrics respectively. Then a Hessian
isometry $\Phi$ from $F_{1}$ to $F_{2}$ is a diffeomorphism on
$\mathbb{R}^{n}\backslash\\{0\\}$ which is an isometry from $g_{1}$ to
$g_{2}$. See Section 4.1 for its basic properties and local version. A linear
isomorphism $\Phi$ on $\mathbb{R}^{n}$ satisfying $F_{1}=F_{2}\circ\Phi$
naturally induces a Hessian isometry when restricted to
$\mathbb{R}^{n}\backslash\\{0\\}$. We call it a linear isometry from $F_{1}$
to $F_{2}$.

As we have seen, linear isometry provides us the hint and shortcut to Theorem
B. Besides, it also helps us prove a special case of Laugwitz Conjecture [29],
which improves Corollary 1.7 in [56] (see Theorem 4.5).

Theorem C. Let $F$ be a Minkowski norm on $\mathbb{R}^{n}$ with $n>2$ induced
by the isoparametric foliation $M_{t}$ on $(S^{n-1}(1),g^{\mathrm{st}})$.
Suppose its Hessian metric $g$ is flat on $\mathbb{R}^{n}\backslash\\{0\\}$,
then $F$ is Euclidean.

The (possibly) nonlinear Hessian isometry between two Minkowski norms induced
by $M_{t}$ is more intriguing. Generally speaking, its complete classification
is a hard problem which involves complicated case-by-case discussion. In this
paper, we only concentrate in a subclass, i.e., we consider the triple
$(F_{1},F_{2},\Phi)$, in which $F_{1}$ and $F_{2}$ are Minkowski norms induced
by $M_{t}$, and the Hessian isometry $\Phi$ from $F_{1}$ to $F_{2}$ preserves
the orientation and fixes the spherical $\xi$-coordinates. There are two ways
to describe this triple.

We may start with the spherical coordinates presentations for
$(F_{1},F_{2},\Phi)$, i.e., $F_{1}=r\sqrt{2f(t)}$, $F_{2}=r\sqrt{2h(\theta)}$
(we use $\theta$ to denote the spherical $t$-coordinate for $F_{2}$), and
$\Phi:(r,t,\xi)\mapsto(\tfrac{rf(t)^{1/2}}{h(\theta(t))^{1/2}},\theta(t),\xi)$,
we find that $(f(t),h(\theta),\theta(t))$ must satisfy the $D_{2d}$-symmetry
and the following ODE system,

$\displaystyle\tfrac{1}{2f(t)}\tfrac{{\rm d}^{2}}{{\rm
d}t^{2}}f(t)-\tfrac{1}{4f(t)^{2}}\left(\tfrac{{\rm d}}{{\rm
d}t}f(t)\right)^{2}+1$ $\displaystyle=$ $\displaystyle\left(\tfrac{{\rm
d}}{{\rm d}t}\theta(t)\right)^{2}\ \left(\tfrac{1}{2h(\theta(t))}\tfrac{{\rm
d}^{2}}{{\rm
d}\theta^{2}}h(\theta(t))-\tfrac{1}{4h(\theta(t))^{2}}\left(\tfrac{{\rm
d}}{{\rm d}\theta}h(\theta(t))\right)^{2}+1\right),\quad\mbox{and}$
$\displaystyle\sin^{2}(t+\tfrac{k\pi}{d})+\tfrac{\cos(t+\tfrac{k\pi}{d})\sin(t+\tfrac{k\pi}{d})}{2f(t)}\tfrac{{\rm
d}}{{\rm d}t}f(t)$ $\displaystyle=$
$\displaystyle\sin^{2}(\theta(t)+\tfrac{k\pi}{d})+\tfrac{\cos(\theta(t)+\tfrac{k\pi}{d})\sin(\theta(t)+\tfrac{k\pi}{d})}{2h(\theta(t))}\tfrac{{\rm
d}}{{\rm d}\theta}h(\theta(t))$ (1.2)

for each $k\in\\{0,\cdots,d-1\\}$.

Alternatively, we may restrict $(F_{1},F_{2},\Phi)$ to any normal plane
$\mathbf{V}$. With $\mathbf{V}$ identified with $\mathbb{R}^{2}$ (see Section
2.4), we get a triple $(\overline{F}_{1},\overline{F}_{2},\overline{\Phi})$
with $D_{2d}$-symmetry, where both $\overline{F}_{i}$ are Minkowski norms on
$\mathbb{R}^{2}$, and $\overline{\Phi}$ is a Hessian isometry between
$\overline{F}_{i}$. In particular, the ODE (1.2) can be interpreted as a
(d)-property, defined by the equality
$g_{x}^{F_{1}}(x^{\prime\prime},x)=g_{\overline{x}}^{F_{2}}(\overline{x}^{\prime\prime},\overline{x})$
for any nonzero $x=x^{\prime}+x^{\prime\prime}$ and
$\Phi(x)=\overline{x}=\overline{x}^{\prime}+\overline{x}^{\prime\prime}$ with
respect to a given orthogonal decomposition
$\mathbb{R}^{n}=\mathbf{V}^{\prime}+\mathbf{V}^{\prime\prime}$. See Definition
5.5 in Section 5.3 and its local version in Section 6.3.

Summarizing Theorem 5.3 and Theorem 5.6, we get the following complete
description for Hessian isometries between two Minkowski norms induced by
$M_{t}$, which preserve the orientation and fix the $\xi$-coordinates.

Theorem D. Let $M_{t}$ be any isoparametric foliation on
$(S^{n-1}(1),g^{\mathrm{st}})$ with $d$ principal curvatures. Then there are
one-to-one correspondences between any two of the following three sets:

  1. (1)

The set of all triples $(F_{1},F_{2},\Phi)$, in which both $F_{i}$ are
Minkowski norms induced by $M_{t}$, and $\Phi$ is a Hessian isometry from
$F_{1}$ to $F_{2}$ which preserves the orientation and fixes the spherical
$\xi$-coordinates;

  2. (2)

The set of all triples $(f(t),h(\theta),\theta(t))$, such that $f(t)$ and
$h(\theta)$ are $D_{2d}$-invariant positive smooth functions on
$\mathbb{R}/(2\mathbb{Z}\pi)$ satisfying the requirement in Theorem A,
$\theta(t)$ is a $D_{2d}$-equivariant orientation preserving diffeomorphism on
$\mathbb{R}/(2\mathbb{Z}\pi)$ fixing each point in $\tfrac{\mathbb{Z}\pi}{d}$,
and the triple is a solution of the ODE system for all
$t\in\mathbb{R}/(2\mathbb{Z}\pi)$, which consists of (1) and (1.2) for all
$k\in\\{0,\cdots,d-1\\}$;

  3. (3)

The set of all triples $(\overline{F}_{1},\overline{F}_{2},\overline{\Phi})$,
in which both $\overline{F}_{i}$ are $D_{2d}$-invariant Minkowski norms on
$\mathbb{R}^{2}$, and $\overline{\Phi}$ is a $D_{2d}$-equivariant orientation
preserving Hessian isometry from $\overline{F}_{1}$ to $\overline{F}_{2}$
which satisfies the (d)-property with respect to the decomposition

$\mathbb{R}^{2}=\mathbf{V}^{\prime}+\mathbf{V}^{\prime\prime}=\mathbb{R}(\cos(-\tfrac{k\pi}{d}),\sin(-\tfrac{k\pi}{d}))+\mathbb{R}(\cos(\tfrac{\pi}{2}-\tfrac{k\pi}{d}),\sin(\tfrac{\pi}{2}-\tfrac{k\pi}{d})))$

for each $k\in\\{0,\cdots,d-1\\}$.

The correspondences from (1) and (3) to (2) are provided by the spherical and
polar coordinates presentations respectively. The correspondence between (1)
and (3) is provided by the restriction to any normal plane $\mathbf{V}$ for
$M_{t}$ and an identification between $\mathbf{V}$ and $\mathbb{R}^{2}$.

Finally, we consider the construction for the Hessian isometry $\Phi$ in
Theorem D.

Legendre transformation (or its composition with a positive scalar
multiplication) provides an important class of (possibly) nonlinear Hessian
isometries [40]. Notice that in this paper we have used the standard inner
product to identify $\mathbb{R}^{n}$ with its dual. So for a Minkowski norm
$F$ induced by $M_{t}$, its dual is also a Minkowski norm on $\mathbb{R}^{n}$
induced by $M_{t}$, and its Legendre transformation preserves the orientation
and fixes the spherical $\xi$-coordinates. Theorem D (or Theorem 5.6) provides
the one-to-one correspondence between Legendre transformations for Minkowski
norms induced by $M_{t}$ and Legendre transformations for $D_{2d}$-invariant
Minkowski norms on $\mathbb{R}^{2}$. See Lemma 6.7 and Theorem 6.8 for the
precise statements.

More examples for the Hessian isometry $\Phi$ in Theorem D can be constructed
by gluing positive scalar multiplications and the compositions of Legendre
transformations and positive scalar multiplications (see Remark 7.3).

On the other hand, when we have $d>2$ for the foliation $M_{t}$, this gluing
construction can exhaust all the wanted $\Phi$. The method for discussing the
ODE system consisting of (1) and (1.2) with $k=0$ (which corresponds to the
(d)-property with $k=0$ in (3) of Theorem D) has been given in [56], which
enables us to locally determine the triple $(f(t),h(\theta),\theta(t))$ around
a generic $t_{0}\in(0,\tfrac{\pi}{d})$. By the assumption $d>2$, (3) in
Theorem D requires essentially more (d)-properties for the triple
$(\overline{F}_{1},\overline{F}_{2},\overline{\Phi})$ (for example, the one
with $k=1$). Applying Lemma 6.6 accordingly, we prove the following theorem
verifying our observation (see Theorem 7.2 for the more precise statement).

Theorem E. Any Hessian isometry between two Minkowski norms induced by an
isoparametric foliation on $(S^{n-1}(1),g^{\mathrm{st}})$ with $d>2$, which
preserves the orientation and fixes the spherical $\xi$-coordinates can be
constructed by gluing positive scalar multiplications and compositions between
the Legendre transformation of $F_{1}$ and positive scalar multiplications. In
particular, it satisfies the (d)-property for any orthogonal decomposition of
$\mathbb{R}^{n}$.

When $d=1$ or $d=2$, Theorem 1.4 and Theorem 1.5 in [56] provide a similar
local description for $\Phi$.

At the end, we remark that most results in this paper for Hessian isometries
are also valid for local Hessian isometries. To avoid iterance and complexity
of terminology and notations, we skip those details.

This paper is organized as following. In Section 2, we introduce the spherical
coordinates and spherical local frame induced by an isoparametric foliation
$M_{t}$ on the unit sphere. In Section 3, we introduce the Minkowski norm
induced by $M_{t}$, and prove Theorem A and Theorem B. In Section 4, we
introduce the notion of Hessian isometry and prove Theorem C. In Section 5, we
study Hessian isometries between two Minkowski norms induced by $M_{t}$ and
prove Theorem D for those which preserve the orientation and fix the spherical
$\xi$-coordinates. In Section 6, we discuss Legendre transformation and
(d)-property. In Section 7, we use the ODE method and (d)-property to provide
the local description for the Hessian isometry $\Phi$ in Theorem D when $d>2$,
and prove Theorem E.

##  2\. Spherical coordinates and spherical local frame induced by an
isoparametric foliation on the unit sphere

###  2.1. Isoparametric function and isoparametric foliation

An isoparametric function on a Riemannian manifold $(M,{g})$ is a smooth
function $p:M\rightarrow\mathbb{R}$ such that it is regular almost everywhere,
and its gradient vector field $\mathrm{grad}\ p$ and its Laplacian $\Delta p$
satisfy

$g(\mathrm{grad}\ p,\mathrm{grad}\ p)=a\circ p\quad\mbox{and}\quad\Delta
p=b\circ p$

for some one-variable functions $a(s)$ and $b(s)$. For each regular value $s$
of $p$, its pre-image $M_{s}=p^{-1}(s)$ is called an isoparametric
hypersurface [13]. We will also use $M_{s}$ to denote the isoparametric
foliation (i.e., the set of all non-empty $M_{s}$). A geodesic is called
normal for (the foliation) $M_{s}$, if it intersects each $M_{s}$
orthogonally.

The isoparametric foliation is called homogeneous if there exists a Lie group
$G$ of isometric actions on $(M,{g})$ such that each $M_{s}$ is a $G$-orbit,
i.e., this isoparametric foliation is induced by the cohomogeneity-one
isometric action of $G$. Indeed, any cohomogeneity-one isometric action can
locally induce an isoparametric foliation.

###  2.2. Isoparametric foliation on a unit sphere

On an Euclidean space $\mathbb{R}^{n}$ with $n\geq 2$, we have the standard
Euclidean inner product $\langle\cdot,\cdot\rangle$, the standard Euclidean
norm $|\cdot|=\langle\cdot,\cdot\rangle^{1/2}$ and the orthonormal linear
coordinates $x=(x_{1},\cdots,x_{n})$. Meanwhile, we have the standard flat
metric on $\mathbb{R}^{n}$, ${g^{\mathrm{st}}}={\rm d}x_{1}^{2}+\cdots+{\rm
d}x_{n}^{2}$. We will also use $g^{\mathrm{st}}$ to denote its restrictions to
submanifolds.

Any isoparametric foliation on $(S^{n-1}(1),{g^{\mathrm{st}}})$ can be related
to an isoparametric function $p:S^{n-1}(1)\rightarrow[-1,1]$, which is the
restriction of a homogeneous polynomial of degree $d\in\\{1,2,3,4,6\\}$ on
$\mathbb{R}^{n}$, where $d$ is the number of principal curvatures. Further
more, $\pm 1$ are the only critical values of $p(\cdot)$. In this foliation,
each $M_{s}=p^{-1}(s)$ with $-1<s<1$ is a closed connected isoparametric
hypersurface, and the two critical sets $M_{\pm 1}$ are the two focal
submanifolds [34].

There are only two subclasses of isoparametric foliations on the unit spheres
[10]. One subclass are those homogeneous ones, which were classified in [24,
50]. The other subclass are of the OT-FKM type [18, 36]. Notice that the OT-
FKM type must have $d=4$, and there is some overlap between the subclasses.

Consider any maximal extended normal geodesic
$\gamma\subset(S^{n-1}(1),g^{\mathrm{st}})$ for $M_{s}$. It is a great circle,
i.e., the intersection between a plane $\mathbf{V}$ passing the origin and
$S^{n-1}(1)$. We will simply call this $\mathbf{V}$ a normal plane for (the
foliation) $M_{s}$, because it coincides with the orthogonal normal complement
of $T_{x}M_{s}$ in $\mathbb{R}^{n}=T_{x}(\mathbb{R}^{n}\backslash\\{0\\})$ for
$x\in\gamma\cap M_{s}$. The intersection $\gamma\cap(M_{-1}\cup M_{1})$ is the
set of a pair of antipodal points when $d=1$, or the vertex set of a regular
$2d$-polygon when $d>1$. The points in $\gamma\cap M_{-1}$ and in $\gamma\cap
M_{1}$ appear alternatively along $\gamma$. Denote
$\mathrm{dist}_{S^{n-1}(1)}(\cdot,\cdot)$ and
$\mathrm{dist}_{\gamma}(\cdot,\cdot)$ the distance functions on
$(S^{n-1}(1),{g^{\mathrm{st}}})$ or $(\gamma,{g^{\mathrm{st}}})$ respectively.
Then we have

$\mathrm{dist}_{S^{n-1}(1)}(M_{-1},M_{1})=\mathrm{dist}_{\gamma}(\gamma\cap
M_{-1},\gamma\cap M_{1})=\tfrac{\pi}{d}.$

For any $s\in(-1,1)$, we have
$c=\mathrm{dist}_{S^{n-1}(1)}(M_{-1},M_{s})\in(0,\tfrac{\pi}{d})$, and

$\gamma\cap M_{s}=\\{x\in\gamma|\mathrm{dist}_{\gamma}(x,\gamma\cap
M_{-1})=c\\}$

contains $2d$ points. The principal curvatures of $M_{s}$, with respect to the
normal direction represented by $\mathrm{grad}\ p$, are exactly
$\cot(c+\tfrac{k\pi}{d})$, $k=0,\cdots,d-1$. The multiplicities of these
principal curvatures are crucial for the classification theory, which has been
extensively studied [1, 17, 46, 47].

###  2.3. Parametrization for an isoparametric foliation on the unit sphere

In later discussion, we will always parametrize an isoparametric foliation on
$(S^{n-1}(1),g^{\mathrm{st}})$ with $d$ principal curvatures as $M_{t}$ with
$t\in[0,\tfrac{\pi}{d}]$, so that for any $x\in M_{t}$ we have

$\mathrm{dist}_{S^{n-1}(1)}(x,M_{0})=\mathrm{dist}_{S^{n-1}(1)}(M_{t},M_{0})=t.$

By this parametrization, $M_{0}$ and $M_{{\pi}/d}$ are the two focal
submanifolds, and all other $M_{t}$ are isoparametric hypersurfaces.
Restricted to each normal geodesic segement realizing the distance from
$M_{0}$ to $M_{t}$ with $0<t\leq\tfrac{\pi}{d}$, $t$ is an
$g^{\mathrm{st}}$-arc length parameter.

Notice that the parameter $t$, when presented as
$t(x)=\mathrm{dist}_{S^{n-1}(1)}(x,M_{0})$, is an isoparametric function on
$(S^{n-1}(1)\backslash(M_{0}\cup M_{\pi/d}),g^{\mathrm{st}})$ for the
foliation $M_{t}$.

###  2.4. Identification between a normal plane and $\mathbb{R}^{2}$ with
$D_{2d}$-action

Later we will frequently use the following identification between
$\mathbb{R}^{2}$ and a normal plane for the isoparametric foliation $M_{t}$ on
$(S^{n-1}(1),g^{\mathrm{st}})$.

Let $\mathbf{V}\subset\mathbb{R}^{n}$ be a normal plane for $M_{t}$, i.e.,
$\gamma=\mathbf{V}\cap S^{n-1}(1)$ is a normal geodesic for $M_{t}$ in
$(S^{n-1}(1),g^{\mathrm{st}})$. We parametrize $\gamma$ as $\gamma(t)$ by its
$g^{\mathrm{st}}$-arc length, and require $\gamma(0)\in M_{0}$. Then
$v_{1}=\gamma(0)$ and $v_{2}=\gamma(\tfrac{\pi}{2})$, when they are viewed as
unit vectors, provide an orthonormal basis for $\mathbf{V}$. We will identify
$\mathbf{V}$ with $\mathbb{R}^{2}$ such that $v_{1}$ and $v_{2}$ are
identified with the standard orthonormal basis vectors $e_{1}=(1,0)$ and
$e_{2}=(0,1)$ respectively.

This identification depends on the choice of $v_{1}$ from $\gamma\cap M_{0}$
and the direction of curve $\gamma(t)$. Changing $v_{1}$ and changing the
direction of $\gamma(t)$ result linear isometries of $\mathbb{R}^{2}$ which
belong to the finite group $D_{2d}$. On the normal plane $\mathbf{V}$ for
$M_{t}$, $D_{2d}$ is the group of all linear isometries which preserves
$\gamma\cap M_{0}$. It is the dihedral group when $d>1$, and $\mathbb{Z}_{2}$
when $d=1$. For $\mathbb{R}^{2}$, $D_{2d}$ is generated by the right
multiplications by

$\left(\begin{array}[]{cc}1&0\\\ 0&-1\\\
\end{array}\right)\quad\mbox{and}\quad\left(\begin{array}[]{cc}\cos\tfrac{2\pi}{d}&\sin\tfrac{2\pi}{d}\\\
-\sin\tfrac{2\pi}{d}&\cos\tfrac{2\pi}{d}\\\ \end{array}\right)$

on row vectors. Alternatively, when we use the polar coordinates
$(r,t)\in\mathbb{R}_{>0}\times\mathbb{R}/(2\mathrm{Z}\pi)$ for
$x=(x_{1},x_{2})=(r\cos t,r\sin t)$ on $\mathbb{R}^{2}$ or
$x=x_{1}v_{1}+x_{2}v_{2}=r\cos t\ v_{1}+r\sin t\ v_{2}$ on $\mathbf{V}$, we
have the corresponding $D_{2d}$-action on the space
$\mathbb{R}/(2\mathrm{Z}\pi)$ of all polar $t$-coordinates, which is generated
by the mappings $t\mapsto-t$ and $t\mapsto t+\tfrac{2\pi}{d}$.

Generally speaking, as long as the subjects we discuss later have the
$D_{2d}$-symmetry (i.e., $D_{2d}$-invariancy or $D_{2d}$-equivariancy), then
it does not depend on the identification when we translate them from
$\mathbf{V}$ to $\mathbb{R}^{2}$ or vice versa.

###  2.5. Spherical coordinates and spherical local frame

Let $M_{t}$ be an isoparametric foliation on $(S^{n-1}(1),g^{\mathrm{st}})$
with $d$ principal curvatures. For any point $x$ in the conic open subset

$C(S^{n-1}(1)\backslash(M_{0}\cup
M_{\pi/d}))=\mathbb{R}_{>0}(S^{n-1}(1)\backslash(M_{0}\cup
M_{\pi/d}))=\mathbb{R}^{n}\backslash(\mathbb{R}_{\geq
0}M_{0}\cup\mathbb{R}_{\geq 0}M_{\pi/d}),$

its spherical coordinates induced by (the foliation) $M_{t}$,
$(r,t,\xi)\in\mathbb{R}_{>0}\times(0,\tfrac{\pi}{d})\times M_{{\pi/2d}}$, is
determined by the following requirements: $r=|x|>0$, $x/|x|\in M_{t}$, and
there exists a normal geodesic segment in $(S^{n-1}(1),g^{\mathrm{st}})$ for
the given foliation, which connects $x$ to $\xi$ without passing the two focal
submanifolds. The mapping from $x$ to its spherical coordinates $(r,t,\xi)$ is
a diffeomorphism between $C(S^{n-1}(1)\backslash(M_{0}\times M_{\pi/d}))$ and
$\mathbb{R}_{>0}\times(0,\tfrac{\pi}{d})\times M_{\pi/2d}$.

Then we construct the local frame in $C(S^{n-1}(1)\backslash(M_{0}\cup
M_{\pi/d}))$ with spherical $\xi$-coordinates contained in some sufficiently
small open subset $U$ of $M_{\pi/2d}$. Here local frame means a set of smooth
tangent vector fields defined on the same open subset, which values at each
point provide a basis of the tangent space.

For the spherical $r$\- and $t$-coordinates induced by $M_{t}$, we have the
tangent vector fields $\partial_{r}$ and $\partial_{t}$. That means,
$\partial_{r}$ generates the rays initiating from the origin, and
$\partial_{t}$ generates the normal geodesics for the foliation
$M_{r,t}=rM_{t}=\\{rx|\forall x\in M_{t}\\}$ on
$(S^{n-1}(r)\backslash(M_{r,0}\cup M_{r,\pi/d}),{g^{\mathrm{st}}})$. Obviously
we have $|\partial_{r}|^{2}={g^{\mathrm{st}}}(\partial_{r},\partial_{r})=1$
and $[\partial_{r},\partial_{t}]=0$. Our convention for parametrizing $M_{t}$
implies $|\partial_{t}|^{2}=r^{2}$.

The other local tangent vector fields, $X_{1},\cdots,X_{n-2}$, are tangent to
the foliation $M_{r,t}$. Firstly, we construct them on $U\subset
M_{{\pi/2d}}$, such that the following are satisfied:

  1. (1)

Each $X_{i}$ is a tangent vector field of constant length on $U$;

  2. (2)

For each $i$, there is a principal curvature value

$\kappa_{i}=\cot(t+\tfrac{k_{i}\pi}{d}),\quad k_{i}\in\\{0,\cdots,d-1\\},$
(2.3)

for $M_{t}$ in $(S^{n+1}(1),{g^{\mathrm{st}}})$, with respect to the normal
direction $\partial_{t}$, such that the value of $X_{i}$ at each point of $U$
is a eigenvector for the eigenvalue $\kappa_{i}$ of the shape operator;

  3. (3)

At each point of $U$, the values of $X_{i}$ for all $1\leq i\leq n-2$ provide
a $g^{\mathrm{st}}$-orthogonal basis for the tangent space of $M_{{\pi/2d}}$.

Notice that any $\xi\in M_{\pi/2d}$ has a sufficient small neighborhood $U$ in
$M_{\pi/2d}$ with no topological obstacle to the above construction of
$X_{1},\cdots,X_{n-2}$. If we ignore the multiplicities, then we have
$\\{k_{1},\cdots,k_{n-2}\\}=\\{0,\cdots,d-1\\}$.

Then we extend each $X_{i}$ such that

$[\partial_{r},X_{i}]=[\partial_{t},X_{i}]=0,\quad\forall 1\leq i\leq n-2.$

Indeed, if $X_{i}$ generates the local diffeomorphisms $\rho_{s}$ on
$M_{{\pi/2d}}$, then after the extension, it generates $\tilde{\rho}_{s}$ with
the spherical coordinates presentation
$\tilde{\rho}_{s}(r,t,\xi)=(r,t,\rho_{s}(\xi))$. At each point,
$X_{1},\cdots,X_{n-2}$ linearly span the tangent space of $M_{r,t}$. So their
brackets $[X_{i},X_{j}]$ are tangent to the foliation $M_{r,t}$ as well. For
simplicity, we denote

$[X_{i},X_{j}]\equiv 0\ \mbox{(mod}\ X_{1},\cdots,X_{n-2}\mbox{)}.$

Along any normal geodesic for $M_{t}$ in $(S^{n-1}(1),{g^{\mathrm{st}}})$,
each $X_{i}$ can be extended to a Jacobi field on the whole great circle,
which vanishes at one of $M_{0}$ and $M_{\pi/d}$. So, $X_{i}/|X_{i}|$ is
parallel along each $t$-curve (i.e. the spherical $r$\- and $\xi$-coordinates
are fixed) for ${g^{\mathrm{st}}}$ on $\mathbb{R}^{n}\backslash\\{0\\}$. By
the positive 1-homogeneity, we may denote

$|X_{i}|^{2}={g^{\mathrm{st}}}(X_{i},X_{i})=r^{2}f_{i}(t)\quad\mbox{with}\quad
f_{i}(t)=a_{i}\sin^{2}(t+\tfrac{k_{i}\pi}{d}),$ (2.4)

where $a_{i}$ is some positive constant and $k_{i}\in\\{0,\cdots,d-1\\}$ is
the integer in (2.3).

Now we have constructed the ${g^{\mathrm{st}}}$-orthogonal local frame
$\\{\partial_{r},\partial_{t},X_{1},\cdots,X_{n-2}\\}$. For simplicity, we
will call it a spherical local frame induced by $M_{t}$. Above discussion can
be summarized as the following lemma.

######  Lemma 2.1.

Let $\\{\partial_{r},\partial_{t},X_{1},\cdots,X_{n-2}\\}$ be a spherical
local frame induced by the isoparametric foliation $M_{t}$ on
$(S^{n-1}(1),g^{\mathrm{st}})$ with $d$ principal curvatures. Then we have the
following:

  1. (1)

The brackets among tangent vector fields in this spherical local frame satisfy

$\displaystyle[\partial_{r},\partial_{t}]=0,\quad[\partial_{r},X_{i}]=[\partial_{t},X_{i}]=0,\,\forall
i,$ $\displaystyle[X_{i},X_{j}]\equiv 0\,\mathrm{(mod}\
X_{1},\cdots,X_{n-2}\mathrm{)},\,\forall i,j.$

  2. (2)

The standard flat metric ${g^{\mathrm{st}}}$ on
$\mathbb{R}^{n}\backslash\\{0\\}$ can be presented as

${g^{\mathrm{st}}}={d}r^{2}+r^{2}{d}t^{2}+r^{2}f_{1}(t)\theta_{1}^{2}+\cdots+r^{2}f_{n-2}(t)\theta_{n-2}^{2},$

where $\\{{\rm d}r,{\rm d}t,\theta_{1},\cdots,\theta_{n-2}\\}$ is the dual
frame for $\\{\partial_{r},\partial_{t},X_{1},\cdots,X_{n-2}\\}$, and
$f_{i}(t)$ is given in (2.4).

Using Lemma 2.1, we can further calculate the Levi-Civita connection of
$(\mathbb{R}^{n}\backslash\\{0\\},{g^{\mathrm{st}}})$.

######  Lemma 2.2.

For the Levi-Civita connection ${\tilde{\nabla}}$ of
$(\mathbb{R}^{n}\backslash\\{0\\},{g^{\mathrm{st}}})$, we have

$\displaystyle{\tilde{\nabla}}_{\partial_{r}}\partial_{r}=0,\quad{\tilde{\nabla}}_{\partial_{r}}\partial_{t}={\tilde{\nabla}}_{\partial_{t}}\partial_{r}=\tfrac{1}{r}\partial_{t},\quad{\tilde{\nabla}}_{\partial_{t}}\partial_{t}=-r\partial_{r},$
$\displaystyle{\tilde{\nabla}}_{\partial_{r}}X_{i}={\tilde{\nabla}}_{X_{i}}\partial_{r}=\tfrac{1}{r}X_{i},\
\forall
i,\quad{\tilde{\nabla}}_{\partial_{t}}X_{i}={\tilde{\nabla}}_{X_{i}}\partial_{t}=\tfrac{1}{2f_{i}(t)}\tfrac{{\rm
d}}{{\rm d}t}f_{i}(t)X_{i},\ \forall i,$
$\displaystyle{\tilde{\nabla}}_{X_{i}}X_{i}\equiv-
rf_{i}(t)\partial_{r}-\tfrac{1}{2}\tfrac{{\rm d}}{{\rm
d}t}f_{i}(t)\partial_{t}\ \mathrm{(mod}\ X_{1},\cdots,X_{n-2}\mathrm{)},\
\forall i,$ $\displaystyle{\tilde{\nabla}}_{X_{i}}X_{j}\equiv 0\
\mathrm{(mod}\ X_{1},\cdots,X_{n-2}\mathrm{)},\ \forall i\neq j.$

##  3\. Minkowski norm induced by an isoparametric foliation

###  3.1. Minkowski norm and Hessian metric

A Minkowski norm $F$ on $\mathbb{R}^{n}$ with $n\geq 2$ is a continuous
function satisfying the following conditions:

  1. (1)

Positiveness and smoothness. The restriction of $F$ to
$\mathbb{R}^{n}\backslash\\{0\\}$ is positive and smooth.

  2. (2)

Positive 1-homogeneity. For any $\lambda\geq 0$ and any $x\in\mathbb{R}^{n}$,
$F(\lambda x)=\lambda F(x)$.

  3. (3)

Strong convexity. For the linear coordinates $x=(x_{1},\cdots,x_{n})$, the
Hessian matrix
$\left(\tfrac{{\partial}^{2}}{{\partial}x_{i}{\partial}x_{j}}E\right)$ for
$E=\tfrac{1}{2}F^{2}$, which is also called the fundamental tensor [2], is
positive definite at any $x\neq 0$.

By its positive 1-homogeneity and strong convexity, the Minkowski norm $F$ is
totally determined by its indicatrix, $S_{F}=\\{x\in\mathbb{R}^{n}|F(x)=1\\}$,
which is a smooth convex sphere surrounding the origin. On
$\mathbb{R}^{n}\backslash\\{0\\}$, the Minkowski norm $F$ determines a
Riemannian metric $g=g(\cdot,\cdot)$, such that for any
$u,v\in\mathbb{R}^{n}=T_{x}(\mathbb{R}^{n}\backslash\\{0\\})$, we have

$g(u,v)=\tfrac{\partial^{2}}{\partial s\partial
t}{}|_{s=t=0}\left(\tfrac{1}{2}F(x+su+tv)^{2}\right)$

at $x\in\mathbb{R}^{n}\backslash\\{0\\}$. We call $g$ the Hessian metric of
$F$ and use the same $g$ to denote its restriction to submanifolds of
$\mathbb{R}^{n}\backslash\\{0\\}$. Sometimes, when the norm $F$ and the
nonzero base vector $x$ need to be specified, we denote it as
$g^{F}_{x}(\cdot,\cdot)$.

For example, a Minkowski norm $F$ is Euclidean if and only if its Hessian
matrices are irrelevant to the choice of
$x\in\mathbb{R}^{n}\backslash\\{0\\}$, i.e., the Hessian metric $g$ can be
viewed as an inner product on $\mathbb{R}^{n}$ which satisfies $F(x)\equiv
g(x,x)^{1/2}$ for every $x\in\mathbb{R}^{n}$. An $(\alpha,\beta)$-norm is of
the form $F=\alpha\varphi(\tfrac{\beta}{\alpha})$, in which $\alpha$ is an
Euclidean norm, $\beta$ is a homogeneous linear function, and $\varphi(s)$ is
some positive one-variable function. An $(\alpha_{1},\alpha_{2})$-norm is of
the form $F(x)=\alpha(x)\varphi(\tfrac{\alpha(x_{1})}{\alpha(x)})$, where
$\alpha$ is an Euclidean norm, $\varphi(s)$ is some one-variable function, and
$x=x_{1}+x_{2}$ is with respect to a fixed $\alpha$-orthogonal decomposition
$\mathbb{R}^{n}=\mathbf{V}_{1}+\mathbf{V}_{2}$.

###  3.2. Induced Minkowski norm and a criterion theorem

Let $M_{t}$ be an isoparametric foliation on $(S^{n-1}(1),g^{\mathrm{st}})$
with $d$ principal curvatures. Then we can construct a Minkowski norm $F$ on
$\mathbb{R}^{n}$, requiring it to be constant on each $M_{t}$. For simplicity,
we call it a Minkowski norm induced by $M_{t}$. Using the spherical
coordinates, we can present it as $F=r\sqrt{2f(t)}$, where $f(t)$ is some
positive function on $[0,\tfrac{\pi}{d}]$.

Let us more closely observe the two features in the norm $F=r\sqrt{2f(t)}$,
the foliation $M_{t}$ and the function $f(t)$.

Firstly, when $M_{t}$ satisfies $d=1$ or $2$. The induced Minkowski norm is an
$(\alpha,\beta)$\- or an $(\alpha_{1},\alpha_{2})$-norm. Indeed, by choosing
suitable inner product, all $(\alpha,\beta)$\- and
$(\alpha_{1},\alpha_{2})$-norms can be induced by an isoparametric foliation
on the unit sphere with $d=1$ or $d=2$.

Secondly, the function $f(t)$ for $F=r\sqrt{2f(t)}$ can be determined by the
restriction of $F$ to any maximally extended normal geodesic for $M_{t}$ in
$(S^{n-1}(1),{g^{\mathrm{st}}})$. So $f(t)$ can be extended to a
$D_{2d}$-invariant positive smooth function for
$t\in\mathbb{R}/(2\pi\mathbb{Z})$, i.e., we always have $f(t)=f(-t)$ and
$f(t)=f(t+\tfrac{2\pi}{d})$. The function $f(t)$ after extension is used in
the polar coordinates presentation $\overline{F}=r\sqrt{2f(t)}$ for the
restriction $\overline{F}=F|_{\mathbf{V}}$ of $F$ to any normal plane
$\mathbf{V}$ for $M_{t}$.

The following criterion theorem answers exactly and explicitly when the formal
expression $F=r\sqrt{2f(t)}$ really defines a Minkowski norm.

######  Theorem 3.1.

Let $M_{t}$ be an isoparametric foliation on $(S^{n-1}(1),{g^{\mathrm{st}}})$
with $d$ principal curvatures. Then the following are equivalent:

  1. (1)

The function $f(t)$ on $[0,\tfrac{\pi}{d}]$ defines a Minkowski norm
$F=r\sqrt{2f(t)}$ on $\mathbb{R}^{n}$ induced by $M_{t}$;

  2. (2)

The function $f(t)$ can be extended to a $D_{2d}$-invariant positive smooth
function on $\mathbb{R}/(2\mathbb{Z}\pi)$ such that

$2f(t)\tfrac{{\rm d}^{2}}{{\rm d}t^{2}}f(t)-\left(\tfrac{{\rm d}}{{\rm
d}t}f(t)\right)^{2}+4f(t)^{2}>0$ (3.5)

is satisfied everywhere;

  3. (3)

The function $f(t)$ after extension (as indicated in (2)) defines a
$D_{2d}$-invariant Minkowski norm on $\mathbb{R}^{2}$ with the polar
coordinates presentation $\overline{F}=r\sqrt{2f(t)}$.

Its proof is postponed to Section 3.4.

The special cases of Theorem 3.1 with $d=1$ or $2$ reprove the following known
results for $(\alpha,\beta)$-norms (see the discussion for Proposition 5 in
[11]) and $(\alpha_{1},\alpha_{2})$-norms (see Theorem 3.2 in [15]).

######  Corollary 3.2.

Let $\alpha$, $\beta$ and $\varphi(s)$ be an Euclidean norm, a nonzero
homogeneous linear function on $\mathbb{R}^{n}$ with $n\geq 2$, and a positive
one-variable function respectively. Then
$F=\alpha\varphi(\tfrac{\beta}{\alpha})$ defines a Minkowski norm if and only
if $\varphi(s)$ is a positive smooth function on $[-b,b]$, where $b$ is the
$\alpha$-norm of $\beta$, and $\varphi(s)$ satisfies

$\varphi(s)-s\tfrac{{\rm d}}{{\rm d}s}\varphi(s)+(b^{2}-|s|^{2})\tfrac{{\rm
d}^{2}}{{\rm d}s^{2}}\varphi(s)>0$ (3.6)

on $[-b,b]$.

Proof. We first prove Corollary 3.2 when $n>2$.

Using the suitable $\alpha$-orthonormal coordinates $(x_{1},\cdots,x_{n})$ and
the corresponding spherical $r$\- and $t$-coordinates, the expression
$F=\alpha\varphi(\tfrac{\beta}{\alpha})$ can be changed to $F=r\varphi(b\cos
t)=r\sqrt{2f(t)}$ with $f(t)=\tfrac{1}{2}\varphi(b\cos t)^{2}$ and
$t\in[0,\pi]$. Direct calculation shows

$2f(t)\tfrac{{\rm d}^{2}}{{\rm d}t^{2}}f(t)-\left(\tfrac{{\rm d}}{{\rm
d}t}f(t)\right)^{2}+4f(t)^{2}=\varphi(s)^{3}\ \left(\varphi(s)-s\tfrac{{\rm
d}}{{\rm d}s}\varphi(s)+(b^{2}-|s|^{2})\tfrac{{\rm d}^{2}}{{\rm
d}s^{2}}\varphi(s)\right),$ (3.7)

where $s=b\cos t$.

Assume $F=\alpha\varphi(\tfrac{\beta}{\alpha})$ defines a Minkowski norm, then
by the equivalence between (1) and (2) in Theorem 3.1 for $d=1$, $f(t)$ is a
smooth even function around $t=0$. By L’Hospital Rule and the theory for
implicit function, we can find a function $\psi(s)$ which is smooth at $s=0$,
such that $f(t)=\psi(b^{2}\sin^{2}t)$ around $t=0$. Then
$\varphi(s)=\psi(b^{2}-s^{2})$ is smooth at $s=b$. The smoothness of
$\varphi(s)$ at $s=-b$ can be similarly verified. Checking the other claims
for $\varphi(s)$ in Corollary 3.2 are easy routines.

Assume $\varphi(s)$ satisfies the requirements in Corollary 3.2, then
obviously $f(t)=\tfrac{1}{2}\varphi(b\cos t)^{2}$ is a positive smooth
function on $\mathbb{R}$ which satisfies $f(t)=f(-t)$ and $f(t)=f(t+2\pi)$,
i.e. $f(t)$ is a function on $\mathbb{R}/(2\mathbb{Z}\pi)$ which is invariant
with respect to the action of $D_{2}=\mathbb{Z}_{2}$. The inequality (3.5)
follows immediately after (3.7) and (3.6). Finally, the equivalent between (1)
and (2) in Theorem 3.1 for $d=1$ tells us
$F=\alpha\varphi(\tfrac{\beta}{\alpha})$ is a Minkowski norm.

To summarize, above argument proves Corollary 3.2 when $n>2$. When $n=2$, we
can use the equivalence between (2) and (3) in Theorem 3.1 and similar
argument to prove this corollary.

######  Corollary 3.3.

Let $\alpha$ be an Euclidean norm on $\mathbb{R}^{n}$ with $n\geq 2$,
$\mathbb{R}^{n}=\mathbf{V}^{\prime}+\mathbf{V}^{\prime\prime}$ an
$\alpha$-orthogonal decomposition with $0<\dim\mathbf{V}^{\prime}=m<n$, and
$\varphi(s)$ some positive function on $[0,1]$. Then
$F(x)=\alpha\varphi(\tfrac{\alpha(x_{1})}{\alpha(x)})$ defines a Minkowski
norm if and only if both $\varphi(s)$ and $\psi(s)=\varphi(\sqrt{1-s^{2}})$
are smooth functions on $[0,1]$, and the inequality

$\varphi(s)-s\tfrac{{\rm d}}{{\rm d}s}\varphi(s)+(1-|s|^{2})\tfrac{{\rm
d}^{2}}{{\rm d}s^{2}}\varphi(s)>0$

is satisfied everywhere.

In the proof of Corollary 3.3, we need to apply Theorem 3.1 for $d=2$ to
discuss the case $2\leq m\leq n-2$. As the argument for each case is very
similar to that in the proof of Corollary 3.2, we skip the details.

###  3.3. Some calculation for the Hessian metric

The calculation for the Hessian metric $g$ of the Minkowski norm
$F=r\sqrt{2f(t)}$ by a spherical local frame induced by $M_{t}$ is the
foundation for later discussion. It is a useful observation that $g$ is in
fact the second covariant derivative of $E=\tfrac{1}{2}F^{2}=r^{2}f(t)$ with
respect to the Levi-Civita connection ${\tilde{\nabla}}$ on
$(\mathbb{R}^{n}\backslash\\{0\\},{g^{\mathrm{st}}})$, so we have

$g(X,Y)=X\cdot(Y\cdot E)-({\tilde{\nabla}}_{X}Y)\cdot E,$

in which $\cdot$ denotes the directional derivative action of vector fields on
differentiable functions and $\tilde{\nabla}$ is the Levi-Civita connection on
$(\mathbb{R}^{n}\backslash\\{0\\},g^{\mathrm{st}})$. Use Lemma 2.2 and notice
$X_{i}\cdot E=0$, we get

######  Lemma 3.4.

Let $\\{\partial_{r},\partial_{t},X_{1},\cdots,X_{n-2}\\}$ be a spherical
local frame, and $F=r\sqrt{2f(t)}$ a Minkowski norm, induced by the same
isoparametric foliation $M_{t}$ on $(S^{n-1}(1),{g^{\mathrm{st}}})$, then we
have

$\displaystyle g(\partial_{r},\partial_{r})=2f(t),\quad
g(\partial_{t},\partial_{t})=r^{2}\tfrac{{\rm d}^{2}}{{\rm
d}t^{2}}f(t)+2r^{2}f(t),$ $\displaystyle
g(X_{i},X_{i})=r^{2}\left(2f_{i}(t)f(t)+\tfrac{1}{2}\tfrac{{\rm d}}{{\rm
d}t}f_{i}(t)\tfrac{{\rm d}}{{\rm d}t}f(t)\right),\ \forall i,$ $\displaystyle
g(\partial_{r},\partial_{t})=r\tfrac{{\rm d}}{{\rm d}t}f(t),\quad
g(\partial_{r},X_{i})=g(\partial_{t},X_{i})=0,\ \forall i,\quad
g(X_{i},X_{j})=0,\ \forall i\neq j.$

We see from Lemma 3.4 that, though the spherical local frame
$\\{\partial_{r},\partial_{t},X_{1},\cdots,X_{n-2}\\}$ may not be
$g$-orthogonal, it is close, i.e., replacing $\partial_{t}$ with
$T=\partial_{t}-\tfrac{r}{2f(t)}\tfrac{{\rm d}}{{\rm d}t}f(t)\partial_{r}$,
then the frame $\\{\partial_{r},T,X_{1},\cdots,X_{n-2}\\}$ is $g$-orthogonal.
Using Lemma 3.4, the $g$-norm square of $T$ can be easily calculated. To
summarize, we have

######  Lemma 3.5.

The local frame $\\{\partial_{r},T,X_{1},\cdots,X_{n-2}\\}$ is $g$-orthogonal,
in which $T=\partial_{t}-\tfrac{r}{2f(t)}\tfrac{{\rm d}}{{\rm
d}t}f(t)\partial_{r}$ with

$g(T,T)=\tfrac{r^{2}}{2f(t)}\left(2f(t)\tfrac{{\rm d}^{2}}{{\rm
d}t^{2}}f(t)-\left(\tfrac{{\rm d}}{{\rm d}t}f(t)\right)^{2}+4f(t)^{2}\right).$
(3.8)

Let $\mathbf{V}$ be any normal plane for $M_{t}$, which has a nonempty
intersection with the defining domain for
$\\{\partial_{r},\partial_{t},X_{1},\cdots,X_{n-2}\\}$. Then $\partial_{t}$
can be smoothly extended to the one on $\mathbf{V}\backslash\\{0\\}$
corresponding to the polar $t$-coordinate. Since $\partial_{r}$ can be
globally defined on $\mathbb{R}^{n}\backslash\\{0\\}$ (corresponding to the
spherical $r$-coordinate), we see $T=\partial_{t}-\tfrac{r}{2f(t)}\tfrac{{\rm
d}}{{\rm d}t}f(t)\partial_{r}$ can be extended to a smooth tangent vector
field on $\mathbf{V}\backslash\\{0\\}$ which is nonvanishing everywhere. By
the positive 1-homogeneity, the smooth extensions to
$\mathbf{V}\backslash\\{0\\}$ for $X_{1},\cdots,X_{n-2}$ can also be observed.
It is easy but useful to see that the nonzero values of $X_{1},\cdots,X_{n-2}$
provide a basis for $T_{x}M_{r,t}$ when $x\in\mathbf{V}\cap
M_{r,t}=\mathbf{V}\cap rM_{t}$ with every $r>0$ and $t\in[0,\tfrac{\pi}{d}]$.
We summarize these observations as the following lemma.

######  Lemma 3.6.

Let $\mathbf{V}$ be any normal plane for $M_{t}$, which has a nonempty
intersection with the defining domain for
$\\{\partial_{r},\partial_{t},X_{1},\cdots,X_{n-2}\\}$. Then the restriction
of $\\{\partial_{r},\partial_{t},X_{1},\cdots,X_{n-2}\\}$ to this intersection
can be canonically extended to $\mathbf{V}\backslash\\{0\\}$ satisfying the
following:

  1. (1)

$\partial_{t}$ and $T=\partial_{t}-\tfrac{r}{2f(t)}\tfrac{{\rm d}}{{\rm
d}t}f(t)\partial_{r}$ are nonvanishing everywhere on
$\mathbf{V}\backslash\\{0\\}$.

  2. (2)

The nonzero values of $X_{1},\cdots,X_{n-2}$ provide the basis of the tangent
space for $M_{r,t}$ for every $r>0$ and $t\in[0,\tfrac{\pi}{d}]$.

###  3.4. Proof of Theorem 3.1

We will mainly prove the equivalence between (1) and (2) in Theorem 3.1. The
equivalence between (2) and (3) can be easily observed in the midway.

Firstly, we prove the claim in Theorem 3.1 from (2) to (1). We assume that
$f(t)$ has been extended to a $D_{2d}$-invariant positive smooth function on
$\mathbb{R}/(2\mathbb{Z}\pi)$ as claimed in Theorem 3.1, i.e., we have

$\displaystyle f(t)=f(-t),\quad f(t)=f(\tfrac{2\pi}{d}-t),\quad
f(t)=f(t+\tfrac{2\pi}{d}),$ (3.9) $\displaystyle 2f(t)\tfrac{{\rm d}^{2}}{{\rm
d}t^{2}}f(t)-\left(\tfrac{{\rm d}}{{\rm d}t}f(t)\right)^{2}+4f(t)^{2}>0,$
(3.10)

for every $t\in\mathbb{R}/(2\mathbb{Z}\pi)$. Then we prove $F=r\sqrt{2f(t)}$
is a Minkowski norm induced by $M_{t}$. Its positiveness, positive
1-homogeneity, and smoothness on $C(S^{n-1}(1)\backslash(M_{0}\cup
M_{\pi/d}))$ are obvious.

To prove its smoothness at $\mathbb{R}_{>0}M_{0}$, we can argue as following.
By the first equality in (3.9) and an exercise of L’Hospital Rule, there
exists a positive function $\psi(s)$ such that $f(t)=\psi(t^{2})$ and
$\psi(s)$ is smooth at $s=0$. As a function on
$(S^{n-1}(1)\backslash(M_{0}\cup M_{\pi/d}),g^{\mathrm{st}})$, the spherical
$t$-coordinate coincides with the distance
$\mathrm{dist}_{S^{n-1}(1)}(\cdot,M_{0})$. Using the exponential map for the
normal bundle of $M_{0}$ in $(S^{n-1}(1)\backslash
M_{\pi/d},{g^{\mathrm{st}}})$, we see
$t^{2}=\left(\mathrm{dist}_{S^{n-1}(1)}(\cdot,M_{0})\right)^{2}$ can be
smoothly extended to a neighborhood of $M_{0}$ in $S^{n-1}(1)$. So
$F=r\sqrt{2f(t)}=r\sqrt{2\psi(t^{2})}$ is smooth at $\mathbb{R}_{>0}M_{0}$.

Using the second equality in (3.9) and similar argument, we can prove
$F=r\sqrt{2f(t)}$ is smooth at $\mathbb{R}_{>0}M_{\pi/d}$. Then the smoothness
is verified. Since we have already observed the positiveness, smoothness and
positive 1-homogeneity for $F$, we see its indicatrix $S_{F}$ is a smooth
sphere surrounding the origin.

Now we verify the strong convexity, which is the most essential part of the
proof. Let $\mathbf{V}$ be a normal plane for $M_{t}$. We consider any
spherical local frame $\\{\partial_{r},\partial_{t},X_{1},\cdots,X_{n-2}\\}$
induced by $M_{t}$, which defining domain has a nonempty intersection with
$\mathbf{V}$. Denote $\overline{F}=F|_{\mathbf{V}}$, then we have the polar
coordinates presentation $\overline{F}=r\sqrt{2f(t)}$.

Using $\\{\partial_{r},\partial_{t},X_{1},\cdots,X_{n-2}\\}$ and
$T=\partial_{t}-\tfrac{r}{2f(t)}\tfrac{{\rm d}}{{\rm d}t}f(t)\partial_{r}$,
the Hessian $g(\cdot,\cdot)$ of $E=\tfrac{1}{2}F^{2}$ has the same
presentations as in Lemma 3.4 and Lemma 3.5. For simplicity, we denote
$g(X,Y)$ for $X,Y\in\\{\partial_{r},T,X_{1},\cdots,X_{n-2}\\}$ as
$g_{\alpha\beta}$, where $\alpha$ and $\beta$ are the indices in the ordered
set $\\{r,T,1,\cdots,n-2\\}$. For example $g_{TT}=g(T,T)$,
$g_{ri}=g(\partial_{r},X_{i})$, etc.. Then

$(g_{\alpha\beta})=\mathrm{diag}(g_{rr},g_{TT},g_{11},\cdots,g_{n-2,n-2})$

is a diagonal matrix. By Lemma 3.6, when we restrict our discussion for
$(g_{\alpha\beta})$ to $\mathbf{V}$, the notion of $g_{\alpha\beta}$ can be
smoothly extended everywhere on $\mathbf{V}\backslash\\{0\\}$.

From Lemma 3.4, we know $g_{rr}=2f(t)>0$. By Lemma 3.5, the left side of the
inequality (3.10) coincides with $\tfrac{2f(t)}{r^{2}}\ g_{TT}$. So (3.10)
implies the left up $2\times 2$-block $\mathrm{diag}(g_{rr},g_{TT})$ in the
Hessian matrix $(g_{\alpha\beta})$ is positive definite.

Before we go on with the discuss the other diagonal entries $g_{ii}$ in
$(g_{\alpha\beta})$, we digress to prove $\overline{F}=F|_{\mathbf{V}}$ is a
Minkowski norm on $\mathbf{V}$. Its strong convexity is the only nontrivial
issue for us to consider. Since the polar coordinates presentation
$\overline{F}=r\sqrt{2f(t)}$ looks the same as the spherical coordinates
presentation for $F$, the Hessian matrix of $\overline{F}$ coincides with the
left up $2\times 2$-block in $(g_{\alpha\beta})$. We have seen its positive
definiteness in the previous discussion, so $\overline{F}$ is strongly convex.
Indeed, this simple observation proves the equivalence between (2) and (3) in
Theorem 3.1.

For the convenience when discussing $g_{ii}$, we denote
$N_{t}=S_{F}\cap\mathbb{R}_{>0}M_{t}$ the foliation on $S_{F}$ induced by
$M_{t}$. We parametrize $S_{\overline{F}}=\mathbf{V}\cap S_{F}$ as $c(t)$ with
$t\in\mathbb{R}/(2\mathbb{Z}\pi)$, equivariantly with respect to the action of
$D_{2d}$, such that $c(t)\in\mathbf{V}\cap N_{t}$ for all
$t\in[0,\tfrac{\pi}{d}]$. Then $\gamma(t)=\tfrac{c(t)}{|c(t)|}$ is a maximal
normal geodesic for $M_{t}$ on $(S^{n-1}(1),g^{\mathrm{st}})$ parametrized by
its $g^{\mathrm{st}}$-arc length with $\gamma(0)\in M_{0}$.

By Lemma 3.4, we have for each $1\leq i\leq n-2$,

$\displaystyle g_{ii}$ $\displaystyle=$ $\displaystyle
r^{2}\left(2f_{i}(t)f(t)+\tfrac{1}{2}\tfrac{{\rm d}}{{\rm
d}t}f_{i}(t)\tfrac{{\rm d}}{{\rm d}t}f(t)\right)$ (3.11) $\displaystyle=$
$\displaystyle\tfrac{a_{i}}{(2f(t))^{1/2}}\ \sin(t+\tfrac{k_{i}\pi}{d})\
\left({\sin(t+\tfrac{k_{i}\pi}{d})}(2f(t))^{1/2}+\tfrac{\cos(t+\tfrac{k_{i}\pi}{d})}{(2f(t))^{1/2}}\tfrac{{\rm
d}}{{\rm d}t}f(t)\right),$

at $x=c(t)$ with $t\in[0,\tfrac{\pi}{d}]$. Here the function
$f_{i}(t)=a_{i}\sin^{2}(t+\tfrac{k_{i}\pi}{d})$ with positive constant $a_{i}$
and $k_{i}\in\\{0,\cdots,n-1\\}$ is given in (2.4).

The factor $\sin(t+\tfrac{k_{i}\pi}{d})$ in the right side of (3.11) is non-
negative for $t\in[0,\tfrac{\pi}{d}]$ and $k_{i}\in\\{0,\cdots,d-1\\}$, and it
vanishes if and only if

$\mbox{either}\quad(t,k_{i})=(0,0)\quad\mbox{or}\quad(t,k_{i})=(\tfrac{\pi}{d},d-1).$
(3.12)

The factor
${\sin(t+\tfrac{k_{i}\pi}{d})}(2f(t))^{1/2}+\tfrac{\cos(t+\tfrac{k_{i}\pi}{d})}{(2f(t))^{1/2}}\tfrac{{\rm
d}}{{\rm d}t}f(t)$ coincides with
$g^{\overline{F}}_{c(t)}(c(t),\gamma(\tfrac{\pi}{2}-\tfrac{k_{i}\pi}{d}))$,
i.e., the derivative of $\overline{E}=\tfrac{1}{2}\overline{F}^{2}=r^{2}f(t)$
in the direction of $\gamma(\tfrac{\pi}{2}-\tfrac{k_{i}\pi}{d})$ at $x=c(t)$,
with $k_{i}\in\\{0,\cdots,d-1\\}$.

We denote $s(t)\in[\tfrac{\pi}{2},\tfrac{3\pi}{2}]$ for
$t\in[0,\tfrac{\pi}{d}]$ such that
$g^{\overline{F}}_{c(t)}(c(t),\gamma(s(t)-{\pi}))=g^{\overline{F}}_{c(t)}(c(t),\gamma(s(t)))=0$
and $g^{\overline{F}}_{c(t)}(c(t),\gamma(s))>0$ for all $s\in(s(t)-\pi,s(t))$.
By the first two equalities of (3.9) from the $D_{2d}$-invariancy, we see
$s(0)=\tfrac{\pi}{2}$ and $s(\tfrac{\pi}{d})=\tfrac{\pi}{d}+\tfrac{\pi}{2}$.
When $t=0$, the interval $(\tfrac{\pi}{2}-\tfrac{(d-1)\pi}{d},\tfrac{\pi}{2})$
is contained in $(s(0)-\pi,s(0))$ with the same right end points. By the
strong convexity of $\overline{F}$, $s(t)$ is a strictly increasing continuous
function on $[0,\tfrac{\pi}{d}]$. So
$(\tfrac{\pi}{2}-\tfrac{(d-1)\pi}{d},\tfrac{\pi}{2})$ stays in the moving
interval $(s(t)-\pi,s(t))$ with $t$ increasing, until $t$ reaches
$\tfrac{\pi}{d}$, and the two intervals have the same left end points.

This observation proves

$g^{\overline{F}}_{c(t)}(c(t),\gamma(s))\geq 0,\quad\forall
t\in[0,\tfrac{\pi}{d}],\ \forall
s\in[\tfrac{\pi}{2}-\tfrac{(d-1)\pi}{d},\tfrac{\pi}{2}],$

and the equality happens if and only if

$\mbox{either}\quad(t,s)=(0,\tfrac{\pi}{2})\quad\mbox{or}\quad(t,s)=(\tfrac{\pi}{d},\tfrac{\pi}{d}-\tfrac{\pi}{2}).$

So
${\sin(t+\tfrac{k_{i}\pi}{d})}(2f(t))^{1/2}+\tfrac{\cos(t+\tfrac{k_{i}\pi}{d})}{(2f(t))^{1/2}}\tfrac{{\rm
d}}{{\rm d}t}f(t)$ is a non-negative factor in $g_{ii}$ as well, and it
vanishes if and only if (3.12) happens.

Summarizing above observations and using Lemma 3.5, Lemma 3.6 and the
$D_{2d}$-symmetry, we see $g_{ii}$ is strictly positive at $x\in
S_{\overline{F}}\backslash(N_{0}\cup N_{\pi/d})$, and $g_{ii}$ vanishes at
$x\in S_{\overline{F}}\cap(N_{0}\cup N_{\pi}/d)$ if and only if $X_{i}$
vanishes there. With the normal plane $\mathbf{V}$ for $M_{t}$ changing
arbitrarily, we see $(g_{\alpha\beta})$ is positive definite on
$S_{F}\backslash(N_{0}\cup N_{\pi/d})$.

For $x\in N_{0}$, Lemma 3.5 and Lemma 3.6 tell us that
$T_{x}(\mathbb{R}_{>0}N_{0})=T_{x/|x|}(\mathbb{R}_{>0}M_{0})\subset\mathbb{R}^{n}$
has a $g$-orthogonal basis consisting of $\partial_{r}$ and the all nonzero
values of $X_{i}$ at $x$. So the restriction of $g$ to
$T_{x}(\mathbb{R}_{>0}N_{0})$ is positive definite. On the other hand, for any
nonzero vector $v$ in the $g$-orthogonal complement $\mathbf{V}^{\prime}$ of
$T_{x}(\mathbb{R}_{>0}N_{0})$ in $T_{x}(\mathbb{R}^{n}\backslash\\{0\\})$, $x$
and $v$ linearly span a normal plane for $M_{t}$, and up to a positive scalar
change, $v$ coincides with the values at $x$ for the vector field $T$ on
$\mathbf{V}\backslash\\{0\\}$. By Lemma 3.5 and (3.10), we see $g(v,v)>0$. So
$g$ is positive definite on $\mathbf{V}^{\prime}\subset
T_{x}(\mathbb{R}^{n}\backslash\\{0\\})$ as well.

This argument proves that $g$ is positive definite on $N_{0}$. Similar
argument proves the $g$ is positive definite on $N_{\pi/d}$. By the positive
1-homogeneity, the proof for the strong convexity is done.

To summarize, above argument proves the claim in Theorem 3.1 from (2) to (1).

Nextly, we prove the claim in Theorem 3.1 from (1) to (2), i.e., we assume
$F=r\sqrt{2f(t)}$ is a Minkowski norm and prove the properties of $f(t)$
claimed in Theorem 3.1. We just need to discuss the restriction of $F$ to any
normal plane for $M_{t}$, then we easily see (1) implies (3). The equivalence
between (2) and (3) has been observed in the midway, so the claim from (1) to
(2) is proved.

###  3.5. Foliation on $S_{F}$ induced by $M_{t}$

In Section 3.4, we have mentioned the foliation
$N_{t}=S_{F}\cap\mathbb{R}_{>0}M_{t}$ on $S_{F}$, which is induced by $M_{t}$
on $S^{n-1}(1)$. Now we prove the following theorem.

######  Theorem 3.7.

Let $F=r\sqrt{2f(t)}$ be a Minkowski norm induced by the isoparametric
foliation $M_{t}$ on the unit sphere $(S^{n-1}(1),g^{\mathrm{st}})$ and $g$
its Hessian metric. Then the foliation $N_{t}=S_{F}\cap\mathbb{R}_{>0}M_{t}$
on $(S_{F},g)$ is isoparametric.

Proof. Denote $d$ the number of principal curvatures for the foliation $M_{t}$
on $(S^{n-1}(1),g^{\mathrm{st}})$. The spherical $t$-coordinate can be viewed
as a regular smooth function on the conic open subset
$C(S^{n-1}(1)\backslash(M_{0}\cup M_{\pi/d})=C(S_{F}\backslash(N_{0}\cup
N_{\pi/d}))$, which is still denoted as $t$. Its level sets provides the
foliation $N_{t}$. We will first prove the function
$t|_{S_{F}\backslash(N_{0}\cup N_{\pi/d})}$ is isoparametric on
$(S_{F}\backslash(N_{0}\cup N_{\pi/d}),g)$, where $g$ is the Hessian metric of
$F=r\sqrt{2f(t)}$.

Let $\\{\partial_{r},\partial_{t},X_{1},\cdots,X_{n-2}\\}$ be any spherical
local frame induced by $M_{t}$ and denote
$T=\partial_{t}-\tfrac{r}{2f(t)}\tfrac{{\rm d}}{{\rm d}t}f(t)\partial_{r}$.
Since $\partial_{t}\cdot t=1$ and $\partial_{r}\cdot t=X_{i}\cdot t=0$, by
Lemma 3.5, the gradient field $\mathrm{grad}^{E}t$ on
$C(S_{F}\backslash(N_{0}\cup N_{\pi/d})$ is

$\mathrm{grad}^{E}t=\left(\tfrac{2f(t)}{r^{2}\left(4f(t)^{2}-\left(\tfrac{{\rm
d}}{{\rm d}t}f(t)\right)^{2}+2f(t)\tfrac{{\rm d}^{2}}{{\rm
d}t^{2}}f(t)\right)}\right)\ T,$

and its pointwise $g$-norm square is

$g(\mathrm{grad}^{E}t,\mathrm{grad}^{E}t)={\tfrac{2f(t)}{r^{2}\left(4f(t)^{2}-\left(\tfrac{{\rm
d}}{{\rm d}t}f(t)\right)^{2}+2f(t)\tfrac{{\rm d}^{2}}{{\rm
d}t^{2}}f(t)\right)}}.$

If restricted to $S_{F}\backslash(N_{0}\cup N_{\pi/d})$, where
$r^{2}=(2f(t))^{-1}$,

$g(\mathrm{grad}^{E}t,\mathrm{grad}^{E}t)|_{S_{F}\backslash(N_{0}\cup
N_{\pi/d})}=\tfrac{4f(t)^{2}}{{4f(t)^{2}-\left(\tfrac{{\rm d}}{{\rm
d}t}f(t)\right)^{2}+2f(t)\tfrac{{\rm d}^{2}}{{\rm d}t^{2}}f(t)}},$

is a function of $t$.

Denote $\mathrm{Hess}(\cdot,\cdot)$ the Hessian with respect to $g$ on
$\mathbb{R}^{n}\backslash\\{0\\}$, then we have

$\mathrm{Hess}(X,X)t=X\cdot(X\cdot t)-(\nabla_{X}X)\cdot t=-(\nabla_{X}X)\cdot
t,$

for each $X\in\\{\partial_{r},T,X_{1},\cdots,X_{n-2}\\}$. Here $\nabla$ is the
Levi-Civita connection of $(\mathbb{R}^{n}\backslash\\{0\\},g)$. So the
Laplacian $\Delta^{E}t$ on $(C(S_{F}\backslash(N_{0}\cup N_{\pi/d}),g)$ can be
presented as

$\displaystyle\Delta^{E}t$ $\displaystyle=$
$\displaystyle\tfrac{\mathrm{Hess}(\partial_{r},\partial_{r})t}{g(\partial_{r},\partial_{r})}+\tfrac{\mathrm{Hess}(T,T)t}{g(T,T)}+\sum_{i=1}^{n-2}\tfrac{\mathrm{Hess}(X_{i},X_{i})t}{g(X_{i},X_{i})}$
(3.13) $\displaystyle=$
$\displaystyle-\tfrac{(\nabla_{\partial_{r}}{\partial_{r}})\cdot
t}{g(\partial_{r},\partial_{r})}-\tfrac{(\nabla_{T}T)\cdot
t}{g(T,T)}-\sum_{i=1}^{n-2}\tfrac{(\nabla_{X_{i}}X_{i})\cdot
t}{g(X_{i},X_{i})},$

where the spherical local frame is defined.

Using Lemma 3.4, we collect the following information for $\nabla$:

$\displaystyle\nabla_{\partial_{r}}\partial_{r}$ $\displaystyle=$
$\displaystyle 0,$ (3.14) $\displaystyle\nabla_{T}T$ $\displaystyle=$
$\displaystyle\tfrac{\left(\tfrac{{\rm d}}{{\rm
d}t}f(t)\right)^{3}-2f(t)\tfrac{{\rm d}}{{\rm d}t}f(t)\tfrac{{\rm d}^{2}}{{\rm
d}t^{2}}f(t)+f(t)^{2}\tfrac{{\rm d}^{3}}{{\rm
d}t^{3}}f(t)}{4f(t)^{3}-f(t)\left(\tfrac{{\rm d}}{{\rm
d}t^{2}}f(t)\right)^{2}+2f(t)^{2}\tfrac{{\rm d}^{2}}{{\rm d}t^{2}}f(t)}\
\partial_{t}\ \mbox{(mod}\ \partial_{r},X_{1},\cdots,X_{n-2}\mbox{)},$ (3.15)
$\displaystyle\nabla_{X_{i}}X_{i}$ $\displaystyle\equiv$
$\displaystyle\tfrac{\tfrac{{\rm d}}{{\rm d}t}f_{i}(t)\left(\tfrac{{\rm
d}}{{\rm d}t}f(t)\right)^{2}-4f(t)^{2}\tfrac{{\rm d}}{{\rm
d}t}f_{i}(t)-f(t)\tfrac{{\rm d}}{{\rm d}t}f_{i}(t)\tfrac{{\rm d}^{2}}{{\rm
d}t^{2}}f(t)-f(t)\tfrac{{\rm d}}{{\rm d}t}f(t)\tfrac{{\rm d}^{2}}{{\rm
d}t^{2}}f_{i}(t)}{8f(t)^{2}-2\left(\tfrac{{\rm d}}{{\rm
d}t}f(t)\right)^{2}+4f(t)\tfrac{{\rm d}^{2}}{{\rm d}t^{2}}f(t)}\ \partial_{t}$
(3.16) $\displaystyle\mbox{(mod}\ \partial_{r},X_{1},\cdots,X_{n-2}\mbox{)}.$

Input the formula of $g(X_{i},X_{i})$ in Lemma 3.4, (3.8) in Lemma 3.5, and
(3.14)-(3.16), into (3.13), we see that $\Delta^{E}t$ is the product of
$r^{-2}$ and a function of $t$. In particular, its restriction to $S_{F}$
(where $r^{-2}=2f(t)$) only depends on the values of $t$.

Finally, we consider the gradient
$\mathrm{grad}^{S}(t|_{S^{F}\backslash(N_{0}\cup N_{\pi/d})})$ and the
Laplacian $\Delta^{S}(t|_{S^{F}\backslash(N_{0}\cup N_{\pi/d})})$ on
$(S^{F}\backslash(N_{0}\cup N_{\pi/d}),g)$. Since the function $t$ is constant
along each ray initiating from the origin, $\mathrm{grad}^{E}t$ is tangent to
$S_{F}$, so we have

$\mathrm{grad}^{S}(t|_{S^{F}\backslash(N_{0}\cup
N_{\pi/d})})=(\mathrm{grad}^{E}t)|_{S^{F}\backslash(N_{0}\cup N_{\pi/d})}.$

By (14.3.10) in [4], we also have

$\displaystyle\Delta^{S}(t|_{S^{F}\backslash(N_{0}\cup
N_{\pi/d})})=(\Delta^{E}t)|_{S^{F}\backslash(N_{0}\cup N_{\pi/d})}.$

So both $g(\mathrm{grad}^{S}(t|_{S^{F}\backslash(N_{0}\cup
N_{\pi/d})}),\mathrm{grad}^{S}(t|_{S^{F}\backslash(N_{0}\cup N_{\pi/d})}))$
and $\Delta^{S}(t|_{S^{F}\backslash(N_{0}\cup N_{\pi/d})})$ are functions on
$S_{F}\backslash(N_{0}\cup N_{\pi/d})$ which only depend on $t$.

To summarize, above argument proves that $t$ is isoparametric on
$(S_{F}\backslash(N_{0}\cup N_{\pi/d}),g)$. Then we can choose $\psi(s)$ on
$[0,\tfrac{\pi}{d}]$ which satisfies $\tfrac{{\rm d}}{{\rm d}t}\psi(s)>0$ on
$(0,\tfrac{\pi}{d})$ and can be extended to a smooth function on $\mathbb{R}$
satisfying $\psi(s)=\psi(-s)$ and $\psi(s)=\psi(\tfrac{2\pi}{d}-s)$. Then the
composition $\psi\circ(t|_{S_{F}\backslash(N_{0}\cup N_{\pi/d})})$ can be
extended to a smooth function $p(\cdot)$ on $(S_{F},g)$, which level sets
provide the foliation $N_{t}$. It is easy to verify the isoparametric property
of $p(\cdot)$ from that of $t|_{S_{F}\backslash(N_{0}\cup N_{\pi/d})}$. So the
foliation $N_{t}$ on $(S_{F},g|_{S_{F}})$ is isoparametric.

By Lemma 3.5 and Theorem 3.7, we see
$T=\partial_{t}-\tfrac{r}{2f(t)}\tfrac{{\rm d}}{{\rm d}t}f(t)\partial_{r}$
generates the normal geodesics for $N_{t}$ on $(S_{F},g)$, which have the
spherical coordinates presentations $t\mapsto((2f(t))^{-1/2},t,\xi)$ for any
fixed $\xi\in M_{0}$. So we have the following corollary.

######  Corollary 3.8.

Let $M_{t}$ be an isoparametric foliation on $(S^{n-1}(1),g^{\mathrm{st}})$,
for which we have induced Minkowski norm $F=r\sqrt{2f(t)}$ with the Hessian
metric $g$, and the isoparametric foliation
$N_{t}=S_{F}\cap\mathbb{R}_{>0}M_{t}$ on $(S_{F},g)$. Then the intersection
with $S_{F}$ provides a one-to-one correspondence between the set of all
normal planes for $M_{t}$ and the set of all unparametrized maximally extended
normal geodesics for $N_{t}$ in $(S_{F},g)$.

##  4\. Hessian isometry and Laugwitz conjecture

###  4.1. Hessian isometry and local Hessian isometry

Let $F_{1}$ and $F_{2}$ be two Minkowski norms on $\mathbb{R}^{n}$ with $n\geq
2$ and denote $g_{1}=g_{1}(\cdot,\cdot)$ and $g_{2}=g_{2}(\cdot,\cdot)$ their
Hessian metrics respectively. A diffeomorphism $\Phi$ on
$\mathbb{R}^{n}\backslash\\{0\\}$ is called a Hessian isometry from $F_{1}$ to
$F_{2}$, if it is an isometry from $g_{1}$ to $g_{2}$.

Since the rays initiating from the origin provide the set of all incomplete
geodesics on $(\mathbb{R}^{n}\backslash\\{0\\},g_{i})$, and the Hessian metric
$g_{i}$ on $\mathbb{R}^{n}\backslash\\{0\\}$ has the presentation $g_{i}=({\rm
d}F_{i})^{2}+F_{i}^{2}(g_{i}|_{S_{F_{i}}})$, we have the following easy lemma.

######  Lemma 4.1.

Any Hessian isometry $\Phi$ from $F_{1}$ to $F_{2}$ maps the indicatrix
$S_{F_{1}}$ to the indicatrix $S_{F_{2}}$, and it is positively 1-homogeneous,
i.e., $\Phi(\lambda x)=\lambda\Phi(x)$ for any $\lambda>0$ and any
$x\in\mathbb{R}^{n}\backslash\\{0\\}$. Conversely, any isometry between
$(S_{F_{i}},g_{i})$ can be uniquely extended to a Hessian isometry from
$F_{1}$ to $F_{2}$.

Besides the global Hessian isometry, a local Hessian isometry can be defined
as an isometric diffeomorphism between two conic open subsets in
$(\mathbb{R}^{n}\backslash\\{0\\},g_{i})$ respectively, satisfying the
positive 1-homogeneity. An analog of Lemma 4.1 is valid for local Hessian
isometries, i.e., the restriction to indicatrix provides a one-to-one
correspondence between local Hessian isometries from $F_{1}$ to $F_{2}$ and
local isometries from $(S_{F_{1}},g_{1})$ to $(S_{F_{2}},g_{2})$.

###  4.2. Linear isometry and two applications

Any linear isomorphism
$\Phi:(\mathbb{R}^{n},F_{1})\rightarrow(\mathbb{R}^{n},F_{2})$ which satisfies
$F_{1}=F_{2}\circ\Phi$ naturally induces a Hessian isometry when it is
restricted to $\mathbb{R}^{n}\backslash\\{0\\}$. For simplicity, we call it a
linear isometry.

Here we propose two applications of linear isometries for the Minkowski norm
$F=r\sqrt{2f(t)}$ induced by an isoparametric foliation $M_{t}$ on
$(S^{n-1}(1),g^{\mathrm{st}})$.

Firstly, when the foliation $M_{t}$ on $(S^{n-1}(1),g^{\mathrm{st}})$ is
homogeneous, i.e., there exists a compact connected Lie subgroup $G$ of
$SO(n)$, such that each $M_{t}$ is a $G$-orbit, then the following lemma shows
us a shortcut to Theorem 3.7.

######  Lemma 4.2.

Let $M_{t}$ be a homogeneous isoparametric foliation on
$(S^{n-1}(1),g^{\mathrm{st}})$, and $F=r\sqrt{2f(t)}$ a Minkowski norm induced
by $M_{t}$. Then the foliation $N_{t}=S_{F}\cap\mathbb{R}_{>0}M_{t}$ on
$(S_{F},g)$ is also a homogeneous isoparametric foliation.

Proof. Let $G$ be the compact connected Lie subgroup of $SO(n)$ such that each
$M_{t}$ is a $G$-orbit. Then the induced Minkowski norm $F=r\sqrt{2f(t)}$ is
$G$-invariant. So the $G$-action on $(\mathbb{R}^{n},F)$ is linearly isometric
and it is of cohomogeneity one when restricted to $(S^{F},g)$. Each
$N_{t}=S_{F}\cap\mathbb{R}_{>0}M_{t}$ is $G$-orbit, and singular $G$-orbits
only appear at the two ends, i.e., $N_{0}$ and $N_{{\pi}/d}$. By the theory of
Riemannian manifold of cohomogeneity one, the corresponding isoparametric
function can be constructed.

Theorem 3.7 is a direct corollary of Lemma 4.2 when we have
$d\in\\{1,2,3,6\\}$ for the principal curvatures of $M_{t}$, and for some
subcases with $d=4$.

Secondly, we prove Laugwitz Conjecture in a special case. Laugwitz conjectured
that if the Hessian metric $g$ of a Minkowski norm $F$ is flat on
$\mathbb{R}^{n}\backslash\\{0\\}$ with $n\geq 3$, then $F$ is Euclidean [29].
When $F$ is reversible, i.e., $F(x)=F(-x)$, $\forall x\in\mathbb{R}^{n}$, or
equivalently, when $F$ is absolutely 1-homogeneous, i.e., $F(\lambda
x)=|\lambda|F(x)$, $\forall\lambda\in\mathbb{R},x\in\mathbb{R}^{n}$, F.
Brickell proved this conjecture by the following theorem [3, 39].

######  Theorem 4.3.

Let $F$ be a Minkowski norm on $\mathbb{R}^{n}$ with $n\geq 3$, satisfying the
reversibility condition, i.e., $F(x)=F(-x)$, $\forall x\in\mathbb{R}^{n}$. If
its Hessian metric $g$ is flat on $\mathbb{R}^{n}\backslash\\{0\\}$, then $F$
is Euclidean.

Recently, we proved the following theorem in [56].

######  Theorem 4.4.

Laugwitz conjecture is true for the class of Minkowski norms which are
invariant with respect to the standard block diagonal $SO(n-1)$-action.

Now, we further strengthen it as following.

######  Theorem 4.5.

Laugwitz conjecture is true for Minkowski norms induced by an isoparametric
foliation on the unit sphere.

Proof. Let $M_{t}$ be an isoparametric foliation on
$(S^{n-1}(1),g^{\mathrm{st}})$ with $n>2$ and $d$ the number of principal
curvatures of $M_{t}$. Suppose $F=r\sqrt{2f(t)}$ is a Minkowski norm induced
by $M_{t}$, with a flat Hessian metric $g$ on
$\mathbb{R}^{n}\backslash\\{0\\}$.

Case 1. We have $d\in\\{2,4,6\\}$. In this case, we only need to prove that
$F$ is reversible, then Theorem 4.5 follows from Theorem 4.3 immediately.

To prove our claim, we consider any $x\in S^{n-1}(1)$. Then there always
exists a normal plane $\mathbf{V}$ for $M_{t}$, which contains $x$. The
antipodal map on $\mathbf{V}$ is contained in $D_{2d}$ when $d$ is even. So
the $D_{2d}$-invariancy of $\overline{F}=F|_{\mathbf{V}}$ implies
$F(x)=F(-x)$.

Case 2. We have $d=1$. This case has already been proved by Theorem 4.4.

Case 3. We have $d=3$. In this case, E. Cartan found the following explicit
construction [6, 7]. We present $\mathbb{R}^{n}$ with $n\in\\{5,8,14,26\\}$ as

$\mathbb{R}^{n}=\mathbb{R}^{2}\oplus\mathbb{F}^{3}=\\{(a,b,x,y,z)|\forall
a,b\in\mathbb{R},\ x,y,z\in\mathbb{F}\\}$

with $\mathbb{F}\in\\{\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}\\}$
respectively, such that the standard Euclidean norm is given by

$|(a,b,x,y,z)|=\sqrt{a^{2}+b^{2}+x\overline{x}+y\overline{y}+z\overline{z}}.$

Then isoparametric function for the foliation $M_{t}$ can be chosen as

$\displaystyle p$ $\displaystyle=$ $\displaystyle
a^{3}-3ab^{2}+\tfrac{3a}{2}(x\overline{x}+y\overline{y}-2z\overline{z})+\tfrac{3\sqrt{3}b}{2}(x\overline{x}-y\overline{y})+\tfrac{3\sqrt{3}}{2}\left((xy)z+\overline{(xy)z}\right),$

where $a^{2}+b^{2}+x\overline{x}+y\overline{y}+z\overline{z}=1$.

Both the standard Euclidean norm and the function $p(\cdot)$ are invariant for
the action of $\Phi(a,b,x,y,z)=(a,b,-x,-y,z)$, So $\Phi$ is a linear isometry
on $(\mathbb{R}^{n},F)$. The fixed point set of $\Phi$ is the Euclidean
subspace $\mathbb{R}^{n^{\prime}}=\\{(a,b,0,0,z)|\forall
a,b\in\mathbb{R},z\in\mathbb{F}\\}$ with dimension $n^{\prime}=3,4,6,10$ when
$n=5,8,14,26$ respectively. Restricted to the unit sphere
$S^{n^{\prime}-1}(1)=\mathbb{R}^{n^{\prime}}\cap S^{n-1}(1)$, where we have
$a^{2}+b^{2}+z\overline{z}=1$ and $x=y=0$, the function $p(\cdot)$ is then
given by

$p|_{S^{n^{\prime}-1}(1)}=a^{3}-3ab^{2}-3az\overline{z}=4a^{3}-3a.$

So the connected components of all $M^{\prime}_{t}=\mathbb{R}^{n^{\prime}}\cap
M_{t}$ provide a homogeneous isoparametric foliation on the unit sphere
$(S^{n^{\prime}-1}(1),g^{\mathrm{st}})$ induced by a standard block diagonal
$SO(n^{\prime}-1)$-action.

Denote the restriction $F^{\prime}=F|_{\mathbb{R}^{n^{\prime}}}$. Then
$F^{\prime}$ is invariant with respect to a standard block diagonal action of
$SO(n^{\prime}-1)$. The Hessian metric $g^{\prime}$ of $F^{\prime}$ coincides
with the restriction $g|_{\mathbb{R}^{n^{\prime}}\backslash\\{0\\}}$. Since
$\mathbb{R}^{n^{\prime}}\backslash\\{0\\}$ is the fixed point set of the
linear isometry $\Phi$ on $(\mathbb{R}^{n}\backslash\\{0\\},g)$, it is totally
geodesic. Because $g$ is flat on $\mathbb{R}^{n}\backslash\\{0\\}$,
$g^{\prime}$ is also flat on $\mathbb{R}^{n^{\prime}}\backslash\\{0\\}$. By
Theorem 4.4, $F^{\prime}$ is Euclidean.

If we use the spherical coordinates
$(r,t,\xi)\in\mathbb{R}_{>0}\times(0,\pi)\times S^{n^{\prime}-2}(1)$ on
$\mathbb{R}^{n^{\prime}}$, such that

$a=r\cos t\quad\mbox{and}\quad(b,z)=r\sin t\ \xi,$

then $F^{\prime}$ has the presentation $F^{\prime}=r\sqrt{2f(t)}$, where
$f(t)=c_{1}+c_{2}\cos 2t$ for some constants $c_{1}$ and $c_{2}$ with
$c_{1}>|c_{2}|$. Meanwhile, $f(t)$ only depends on the values of

$p|_{S^{n^{\prime}-1}(1)}=4a^{3}-3a=4\cos^{3}t-3\cos t=\cos 3t,$

which has a zero derivative at $t=\tfrac{\pi}{3}$. So we have $\tfrac{{\rm
d}}{{\rm d}}f(t)|_{t=\pi/3}=0$, i.e., $c_{2}=0$ and $f(t)$ is a positive
constant function.

Since the same $f(t)$ is also used in the presentation $F=r\sqrt{2f(t)}$, we
see $F$ must be Euclidean as well. The case $d=3$ is proved.

##  5\. Hessian isometry which preserves the orientation and fixes the
spherical $\xi$-coordinates

###  5.1. Notations for the spherical coordinates presentation

Let $M_{t}$ be an isoparametric foliation on $(S^{n-1}(1),g^{\mathrm{st}})$
with $d$ principal curvatures. When we mention the spherical coordinates,
spherical local frame, induced Minkowski norm, etc., they are always referred
to $M_{t}$. Let $F_{1}$ and $F_{2}$ be two induced Minkowski norms on
$\mathbb{R}^{n}$. We denote their indicatrices and Hessian metrics as
$S_{F_{i}}$ and $g_{i}$ respectively. On each $(S_{F_{i}},g_{i})$, we have the
induced isoparametric foliation $N_{i,t}=S_{F_{i}}\cap\mathbb{R}_{>0}M_{t}$.
To distinguish the two Minkowski norms, we use $\theta$ to denote the
spherical $t$-coordinate for $F_{2}$. So we have the spherical coordinates
presentations

$F_{1}=r\sqrt{2f(t)}\quad\mbox{and}\quad F_{2}=r\sqrt{2h(\theta)}.$ (5.17)

Though (5.17) only use the values of $f(t)$ and $h(\theta)$ on
$[0,\tfrac{\pi}{d}]$, as Theorem 3.1 indicates, both functions can and will be
extended to $D_{2d}$-invariant positive smooth functions on
$\mathbb{R}/(2\mathbb{Z}\pi)$, and they satisfy the inequality in Theorem 3.1.

Let $\mathbf{V}$ be any normal plane for $M_{t}$. We parametrize
$\mathbf{V}\cap S^{n-1}(1)$ as $\gamma(t)$ with its $g^{\mathrm{st}}$-arc
length parameter $t\in\mathbb{R}/(2\mathbb{Z}\pi)$ and $\gamma(0)\in M_{0}$.
The normal plane $\mathbf{V}$ can be identified with $\mathbb{R}^{2}$ as
indicated in Section 2.4, so that $v_{1}=\gamma(0)$ and
$v_{2}=\gamma(\tfrac{\pi}{2})$ are mapped to $e_{1}=(1,0)$ and $e_{2}=(0,1)$
respectively. The polar coordinates
$(r,t)\in\mathbb{R}_{>0}\times(\mathbb{R}/(2\mathbb{Z}\pi))$ on $\mathbf{V}$
is determined by

$x=x_{1}v_{1}+x_{2}v_{2}=r\cos t\ v_{1}+r\sin t\ v_{2}$

for any $x\in\mathbf{V}\backslash\\{0\\}$. Each interval
$(\tfrac{k\pi}{d},\tfrac{(k+1)\pi}{d})$ with $k\in\\{0,\cdots,2d-1\\}$ for the
polar $t$-coordinate determines a conic open subset of
$\mathbf{V}\backslash\\{0\\}$, which corresponds to a distinct spherical
$\xi$-coordinate on $\mathbb{R}^{n}$. The restrictions
$\overline{F}_{i}=F_{i}|_{\mathbf{V}}$ have the polar coordinates
presentations $\overline{F}_{1}=r\sqrt{2f(t)}$ and
$\overline{F}_{2}=r\sqrt{2h(\theta)}$ (similarly we use $\theta$ to denote the
polar $t$-coordinates for $\overline{F}_{2}$), where $f(t)$ and $h(\theta)$
are exactly those in (5.17) after extension.

In this section, we discuss a Hessian isometry $\Phi$ from $F_{1}$ to $F_{2}$
which preserves the orientation and fixes the spherical $\xi$-coordiantes,
i.e., it satisfies the following conditions:

  1. (1)

$\Phi$ is an orientation preserving diffeomorphism on
$\mathbb{R}^{n}\backslash\\{0\\}$.

  2. (2)

$\Phi$ preserves the conic open subset $C(S^{n-1}(1)\backslash M_{0}\cup
M_{\pi/d})=\mathbb{R}^{n}\backslash(\mathbb{R}_{\geq
0}M_{0}\cup\mathbb{R}_{\geq 0}M_{{\pi}/{d}})$, and for any $x\in
C(S^{n-1}(1)\backslash M_{0}\cup M_{\pi/d})$, $x$ and $\Phi(x)$ has the same
$\xi$-coordinates.

By Lemma 4.1, we have $\Phi(S_{F_{1}})=S_{F_{2}}$. Since the condition (2)
requires that $\Phi$ fixes the spherical $\xi$-coordinates, our previous
observation indicates $\Phi$ preserves each arbitrarily chosen normal plane
$\mathbf{V}$, i.e., $\overline{\Phi}=\Phi|_{\mathbf{V}}$ is a Hessian isometry
from $\overline{F}_{1}=F_{1}|_{\mathbf{V}}$ to
$\overline{F}_{2}=F_{2}|_{\mathbf{V}}$. Further more, $\Phi$ fixes each point
in $\mathbf{V}\cap(M_{0}\cup M_{\pi/2d})$ and preserves each conic open subset
in $\mathbf{V}\backslash\\{0\\}$ with polar $t$-coordinate in
$(\tfrac{k\pi}{d},\tfrac{(k+1)\pi}{d})$.

More discussion for the restriction to $\mathbf{V}$ is postponed to Section
5.3. Here we only need to apply Corollary 3.8 to notice that $\Phi$ maps each
normal geodesic for the isoparametric foliation
$N_{1,t}=S_{F_{1}}\cap\mathbb{R}_{>0}M_{t}$ on $(S_{F_{1}},g_{1})$ to that for
$N_{2,t}=S_{F_{2}}\cap\mathbb{R}_{>0}M_{t}$ on $(S_{F_{2}},g_{2})$. Meanwhile,
$\Phi$ preserves the foliation $\mathbb{R}_{>0}M_{t}$ on
$\mathbb{R}^{n}\backslash\\{0\\}$. So $\Phi$ maps each
$N_{1,t}=S_{F_{1}}\cap\mathbb{R}_{>0}M_{t}$ to some
$N_{2,\theta(t)}=S_{F_{2}}\cap\mathbb{R}_{>0}M_{\theta(t)}$. To summarize,
$\Phi$ has the following spherical coordinates presentation

$(r,t,\xi)\mapsto(\tfrac{rf(t)^{1/2}}{h(\theta(t))^{1/2}},\theta(t),\xi).$

Since $\Phi$ is an orientation preserving diffeomorphism by the condition (1),
it can be observed immediately from the Jacobi matrix

$\left(\begin{array}[]{ccc}\tfrac{f(t)^{1/2}}{h(\theta(t))^{1/2}}&\tfrac{h(\theta(t))-f(t)\tfrac{{\rm
d}}{{\rm d}t}\theta(t)}{2f(t)(2h(\theta(t)))^{3/2}}&0\\\ 0&\tfrac{{\rm
d}}{{\rm d}t}\theta(t)&0\\\ 0&0&\mathrm{Id}\\\ \end{array}\right).$

for the tangent map $\Phi_{*}$ that $\theta(t)$ is a diffeomorphism on
$(0,\tfrac{\pi}{d})$ with positive derivative everywhere, and $\theta(t)=t$
for $t\in\\{0,\tfrac{\pi}{d}\\}$ by continuity. Indeed, $\theta(t)$ can and
will be extended an orientation preserving diffeomorphism on
$\mathbb{R}/(2\mathbb{Z}\pi)$ with a $D_{2d}$-equivariancy, i.e.,
$\theta(-t)=-\theta(t)$ and
$\theta(t+\tfrac{2\pi}{d})=\theta(t)+\tfrac{2\pi}{d}$, which will appear in
the polar coordinates presentation
$(r,t)\rightarrow(\tfrac{rf(t)^{1/2}}{h(\theta(t))^{1/2}},\theta(t))$ for
$\overline{\Phi}=\Phi|_{\mathbf{V}}$. After the extension, $\theta(t)$ fixes
each point in $\tfrac{\mathbb{Z}\pi}{d}\subset\mathbb{R}/(2\mathbb{Z}\pi)$.

Summarizing above argument, we get the following lemma for the spherical
coordinates presentation for $\Phi$.

######  Lemma 5.1.

Let $F_{1}=r\sqrt{2f(t)}$ and $F_{2}=r\sqrt{2h(\theta)}$ be two Minkowski
norms induced by $M_{t}$. Then any Hessian isometry $\Phi$ from $F_{1}$ to
$F_{2}$ which preserves the orientation and fixes the spherical
$\xi$-coordinates has the spherical coordinates presentation

$(r,t,\xi)\mapsto(\tfrac{rf(t)^{1/2}}{h(\theta(t))^{1/2}},\theta(t),\xi),$
(5.18)

in which $\theta(t)$ is a $D_{2d}$-equivariant orientation preserving
diffeomorphism on $\mathbb{R}/(2\mathbb{Z}\pi)$ and fixes each point in
$\tfrac{\mathbb{Z}\pi}{d}$.

###  5.2. Description by an ODE system

Nextly, we consider a spherical local frame $\\{\partial_{r}$, $\partial_{t}$
(or $\partial_{\theta}$ for $F_{2}$), $X_{1},\cdots,X_{n-2}\\}$, defined in
the conic open subset of $\mathbb{R}^{n}\backslash\\{0\\}$ which only requires
the spherical $\xi$-coordinate to be contained in some open subset in
$M_{\pi/2d}$, i.e., its defining domain is preserved by $\Phi$. We denote
$T_{1}=\partial_{t}-\tfrac{r}{2f(t)}\tfrac{{\rm d}}{{\rm d}t}f(t)\partial_{r}$
and $T_{2}=\partial_{\theta}-\tfrac{r}{2h(\theta)}\tfrac{{\rm d}}{{\rm
d}\theta}h(\theta)\partial_{r}$ which are tangent to $S_{F_{1}}$ and
$S_{F_{2}}$ respectively. Then we have

######  Lemma 5.2.

The tangent map $\Phi_{*}$ for $\Phi$ satisfies

$\displaystyle\Phi_{*}(T_{1})$ $\displaystyle=$ $\displaystyle\tfrac{{\rm
d}}{{\rm d}t}\theta(t)\ T_{2},\quad\mbox{and}$ (5.19)
$\displaystyle\Phi_{*}(X_{i})$ $\displaystyle=$ $\displaystyle
X_{i},\quad\forall 1\leq i\leq n-2.$ (5.20)

Proof. We first prove (5.19). By the positive 1-homogeneity (i.e., Lemma 4.1),
we may restrict our discussion to $S_{F_{1}}$. On $(S_{F_{1}},g_{1})$, $T_{1}$
generates a normal geodesic $c_{1}(t)$ with $t\in(0,\tfrac{\pi}{d})$ for the
isoparametric foliation $N_{1,t}$. Its image $\Phi(c_{1}(t))$ coincides with a
normal geodesic $c_{2}(\theta)$ with $\theta\in(0,\tfrac{\pi}{d})$ for the
isoparametric foliation $N_{2,\theta}$ on $(S_{F_{2}},g_{2})$, i.e., an
integral curve of $T_{2}$, up to a change of parameter $\theta=\theta(t)$. So
we have $\Phi_{*}(T_{1})=\tfrac{{\rm d}}{{\rm d}t}\theta(t)\ T_{2}$.

Then we prove (5.20). Let $s\mapsto((2f(t))^{-1/2},t,\xi(s))$ with any fixed
$t\in(0,\tfrac{\pi}{d})$ be the spherical coordinates presentation for an
integral curve of $X_{i}$ on $S_{F_{1}}$, then its $\Phi$-image has the
spherical coordinates presentation
$s\mapsto((2h(\theta(t)))^{-1/2},\theta(t),\xi(s))$, which is still an
integral curve of $X_{i}$. So we have $\Phi_{*}(X_{i})=X_{i}$, $\forall 1\leq
i\leq n-2$.

Since $\Phi$ is a Hessian isometry, (5.19) implies

$g_{1}(T_{1},T_{1})=g_{2}(\Phi_{*}(T_{1}),\Phi_{*}(T_{1}))=\left(\tfrac{{\rm
d}}{{\rm d}t}\theta(t)\right)^{2}\ g_{2}(T_{2},T_{2}),$ (5.21)

where the left side is evaluated at $x=((2f(t))^{-1/2},t,\xi)\in
S_{F_{1}}\backslash(N_{1,0}\cup N_{1,\pi/d})$ and the right side is evaluated
at $\Phi(x)=((2h(\theta(t)))^{-1/2},\theta(t),\xi)\in
S_{F_{2}}\backslash(N_{2,0}\cup N_{2,\pi/d})$. Using (3.8) in Lemma 3.5, we
get the following ODE,

$\displaystyle\tfrac{1}{2f(t)}\tfrac{{\rm d}^{2}}{{\rm
d}t^{2}}f(t)-\tfrac{1}{4f(t)^{2}}\left(\tfrac{{\rm d}}{{\rm
d}t}f(t)\right)^{2}+1$ (5.22) $\displaystyle=$ $\displaystyle\left(\tfrac{{\rm
d}}{{\rm d}t}\theta(t)\right)^{2}\ \left(\tfrac{1}{2h(\theta(t))}\tfrac{{\rm
d}^{2}}{{\rm
d}\theta^{2}}h(\theta(t))-\tfrac{1}{4h(\theta(t))^{2}}\left(\tfrac{{\rm
d}}{{\rm d}\theta}h(\theta(t))\right)^{2}+1\right),$

for $t\in(0,\tfrac{\pi}{d})$. By continuity and $D_{2d}$-symmetry, (5.22) for
$t\in(0,\tfrac{\pi}{d})$ is equivalent to that for all
$t\in\mathbb{R}/(2\mathbb{Z}\pi)$.

Similarly, (5.20) implies $g_{1}(X_{i},X_{i})=g_{2}(X_{i},X_{i})$ for each $i$
. Using Lemma 3.4, we get the following ODEs,

$f_{i}(t)+\tfrac{1}{4f(t)}\tfrac{{\rm d}}{{\rm d}t}f_{i}(t)\tfrac{{\rm
d}}{{\rm d}t}f(t)=f_{i}(\theta(t))+\tfrac{1}{4h(\theta(t))}\tfrac{{\rm
d}}{{\rm d}\theta}f_{i}(\theta(t))\tfrac{{\rm d}}{{\rm d}\theta}h(\theta(t)),$
(5.23)

for any $1\leq i\leq n-2$ and any $t\in(0,\tfrac{\pi}{d})$. Here
$f_{i}(t)=a_{i}\sin^{2}(t+\tfrac{k_{i}\pi}{d})$ with constants $a_{i}>0$ and
$k_{i}\in\\{0,\cdots,d-1\\}$. Because
$\\{k_{1},\cdots,k_{n-2}\\}=\\{0,\cdots,d-1\\}$, we can reorganize (5.23) as

$\displaystyle\sin^{2}(t+\tfrac{k\pi}{d})+\tfrac{\cos(t+\tfrac{k\pi}{d})\sin(t+\tfrac{k\pi}{d})}{2f(t)}\tfrac{{\rm
d}}{{\rm d}t}f(t)$ (5.24) $\displaystyle=$
$\displaystyle\sin^{2}(\theta(t)+\tfrac{k\pi}{d})+\tfrac{\cos(\theta(t)+\tfrac{k\pi}{d})\sin(\theta(t)+\tfrac{k\pi}{d})}{2h(\theta(t))}\tfrac{{\rm
d}}{{\rm d}\theta}h(\theta(t)),$

for every $t\in(0,\tfrac{\pi}{d})$ and every $k\in\\{0,\cdots,d-1\\}$.

Obviously, the ODEs in (5.24) are satisfied with all $k\in\mathbb{Z}$. Further
more, they are satisfied for all $t\in\mathbb{R}/(2\mathbb{Z}\pi)$ as well. To
prove this claim, we first observe that by continuity, (5.24) for each $k$ is
valid at $t=0$ and $t=\tfrac{\pi}{d}$. Using the properties $f(t)=f(-t)$,
$h(\theta)=h(-\theta)$ and $\theta(-t)=-\theta(t)$, (5.24) with $k=k^{\prime}$
for $t\in[-\tfrac{\pi}{d},0]$ can be deduced from (5.24) with $k=d-k^{\prime}$
for $t\in[0,\tfrac{\pi}{d}]$. Then using the symmetry with respect to
$\mathbb{Z}_{g}\subset D_{g}$, i.e., the properties
$f(t)=f(t+\tfrac{2\pi}{d})$, $h(\theta)=h(\theta+\tfrac{2\pi}{d})$ and
$\theta(t+\tfrac{2\pi}{d})=\theta(t)+\tfrac{2\pi}{d}$, (5.24) with
$k=k^{\prime}$ for
$t\in[\tfrac{(2k^{\prime\prime}-1)\pi}{d},\tfrac{(2k^{\prime\prime}+1)\pi}{d}]$
can be deduced from (5.24) with $k=k^{\prime}+2k^{\prime\prime}$ for
$t\in[-\tfrac{\pi}{d},\tfrac{\pi}{d}]$.

Above argument tells us how to determine the triple
$(f(t),h(\theta),\theta(t))$ from $(F_{1},F_{2},\Phi)$, and more importantly,
list the properties the triple $(f(t),h(\theta),\theta(t))$ must satisfy. Then
we observe how to use the data $(f(t),h(\theta),\theta(t))$ with those
properties to construct the wanted $(F_{1},F_{2},\Phi)$.

When the positive smooth $D_{2d}$-invariant functions $f(t)$ and $h(\theta)$
satisfy the inequality in Theorem 3.1, we can use them construct the induced
Minkowski norms $F_{1}=r\sqrt{2f(t)}$ and $F_{2}=r\sqrt{2h(\theta)}$.

When the $D_{2d}$-equivariant orientation preserving diffeomorphism
$\theta(t)$ on $\mathbb{R}/(2\mathbb{Z}\pi)$ fixes each point in
$\tfrac{\mathbb{Z}\pi}{d}$, we can use it to construct a diffeomorphism $\Phi$
on $C(S^{n-1}(1)\backslash(M_{0}\cup M_{\pi/d}))$ with the spherical
coordinates presentation (5.18), i.e.,
$(r,t,\xi)\mapsto(\tfrac{rf(t)^{1/2}}{h(\theta(t))^{1/2}},\theta(t),\xi)$.
Then obviously $\Phi$ preserves the orientation and fixes the spherical
$\xi$-coordinates. To see it can be extended to a Hessian isometry, we only
need to consider its restriction between the two indicatrices.

Let $\\{\partial_{r}$, $\partial_{t}$ (or $\partial_{\theta}$ for $F_{2}$),
$X_{1},\cdots,X_{n-2}\\}$ be any spherical local frame, and denote
$T_{1}=\partial_{t}-\tfrac{r}{2f(t)}\tfrac{{\rm d}}{{\rm d}t}f(t)\partial_{r}$
and $T_{2}=\partial_{\theta}-\tfrac{r}{2h(\theta)}\tfrac{{\rm d}}{{\rm
d}\theta}h(\theta)\partial_{r}$. Similarly, from (5.18), we see that $\Phi$
preserves the defining domain for this spherical local frame,
$\Phi_{*}(T_{1})=\tfrac{{\rm d}}{{\rm d}t}\theta(t)\ T_{2}$, and
$\Phi_{*}(X_{i})=X_{i}$, $\forall 1\leq i\leq n-2$, as in Lemma 5.2. Then
(5.22) and (5.24) for all $k\in\\{0,\cdots,d-1\\}$ implies

$g_{1}(T_{1},T_{1})=g_{2}(\Phi_{*}(T_{1}),\Phi_{*}(T_{1})),\quad\mbox{and}\quad
g_{1}(X_{i},X_{i})=g_{2}(\Phi_{*}(X_{i}),\Phi_{*}(X_{i})),\ \forall i.$

Since $\\{T_{i},X_{1},\cdots,X_{n-2}\\}$ is $g_{i}$-orthogonal, we see $\Phi$
is an isometry between $(S_{F_{i}}\backslash(N_{i,0}\cup N_{i,{\pi}/d})$,
$g_{i})$.

Then we prove that $\Phi$ can be continuously glued with some scalar
multiplications from $N_{1,t}$ to $N_{2,t}$ for $t=0,\tfrac{\pi}{d}$
respectively. Equivalently, we show the following map,

$\Phi_{1}(x)=\tfrac{\Phi(x)}{|\Phi(x)|},\ \forall x\in
S^{n-1}(1)\backslash(M_{0}\cup M_{\pi/d}),\quad\Phi_{1}(x)=x,\ \forall x\in
M_{0}\cup M_{\pi/d},$

is continuous on $S^{n-1}(1)$. Using the exponential maps for the normal
bundle of the two focal submanifolds $M_{0}$ and $M_{\pi/d}$ in
$(S^{n-1}(1),g^{\mathrm{st}})$, we see $S^{n-1}(1)$ has a topological basis
consisting of the following open subsets:

  1. (1)

Open subsets $U$ of $S^{n-1}(1)\backslash(M_{0}\cup M_{\pi/d})$;

  2. (2)

$U_{U^{\prime},c}$ parametrized by the open subset $U^{\prime}\subset M_{0}$
and the real number $c\in(0,\tfrac{\pi}{d})$, which contains every point $x\in
S^{n-1}(1)$ which satisfies $\mathrm{dist}_{S^{n-1}(1)}(x,M_{0})<c$ and is
contained in some normal geodesic for $M_{t}$ in
$(S^{n-1}(1),g^{\mathrm{st}})$ passing $U^{\prime}$.

  3. (3)

$U_{U^{\prime\prime},c}$ parametrized by the open subset
$U^{\prime\prime}\subset M_{\pi/d}$ and the real number
$c\in(0,\tfrac{\pi}{d})$, which contains every point $x\in S^{n-1}(1)$ which
satisfies $\mathrm{dist}_{S^{n-1}(1)}(x,M_{\pi/d})<c$ and is contained in some
normal geodesic for $M_{t}$ in $(S^{n-1}(1),g^{\mathrm{st}})$ passing
$U^{\prime\prime}$.

Restricted to $S^{n-1}(1)\backslash(M_{0}\cup M_{\pi/d})$, $\Phi_{1}$ is a
diffeomorphism with the spherical coordinates presentation
$(1,t,\xi)\mapsto(1,\theta(t),\xi)$. So for any open subset $U\in
S^{n-1}(1)\backslash(M_{0}\cup M_{\pi/d})$, $\Phi_{1}^{-1}(U)$ is still an
open subset in $S^{n-1}(1)\backslash(M_{0}\cup M_{\pi/d})$. By the spherical
coordinates presentation for $\Phi_{1}$, we see
$\Phi_{1}^{-1}({U}_{c,U^{\prime}})={U}_{\theta^{-1}(c),U^{\prime}}$ and
$\Phi_{1}^{-1}({U}_{c,U^{\prime\prime}})={U}_{c_{1},U^{\prime\prime}}$ with
$c_{1}=\tfrac{\pi}{d}-\theta^{-1}(\tfrac{\pi}{d}-c)$, for any
$c\in(0,\tfrac{\pi}{d})$, any open subset $U^{\prime}\subset M_{0}$ and any
open subset $U^{\prime\prime}\subset M_{\pi/d}$. Here $t$ and $\theta(t)$ are
viewed as numbers in $(0,\tfrac{\pi}{d})$. So $\Phi_{1}$ is continuous on
$S^{n-1}(1)$. Meanwhile, we see $\Phi$ can be continuously extended to a
homomorphism from $S_{F_{1}}$ to $S_{F_{2}}$.

Because $\Phi$ is isometric when restricted to
$(S_{F_{i}}\backslash(N_{i,0}\cup N_{i,\pi/d}),g_{i})$, it is still an
isometry after the extension. Finally, by Lemma 4.1, we can use the positive
1-homogeneity to further extend $\Phi$ to a Hessian isometry between $F_{1}$
and $F_{2}$.

Above discussion tells us the triple $(F_{1},F_{2},\Phi)$ can be constructed
from the triple $(f(t),h(\theta)$, $\theta(t))$. So we have the one-to-one
correspondence in the following theorem.

######  Theorem 5.3.

Let $M_{t}$ be an isoparametric foliation on $(S^{n-1}(1),g^{\mathrm{st}})$
with $d$ principal curvatures. Then the spherical coordinates presentations,
$F_{1}=r\sqrt{2f(t)}$, $F_{2}=r\sqrt{2h(\theta)}$ and
$\Phi:(r,t,\xi)\mapsto(\tfrac{rf(t)^{1/2}}{h(\theta(t))^{1/2}},\theta(t),\xi)$,
provide the one-to-one correspondence between the set of all triples
$(F_{1},F_{2},\Phi)$ satisfying the following:

  1. (1)

$F_{1}$ and $F_{2}$ are Minkowski norms on $\mathbb{R}^{n}$ induced by
$M_{t}$;

  2. (2)

$\Phi$ is a Hessian isometry from $F_{1}$ to $F_{2}$ which preserves the
orientation and fixes the spherical $\xi$-coordinates.

and the set of all triples $(f(t),h(\theta),\theta(t))$ satisfying the
following:

  1. (1)

$f(t)$ and $h(\theta)$ are $D_{2d}$-invariant positive smooth functions on
$\mathbb{R}/(2\mathbb{Z}\pi)$ satisfying the inequalities

$\displaystyle 2f(t)\tfrac{{\rm d}^{2}}{{\rm d}t^{2}}f(t)-\left(\tfrac{{\rm
d}}{{\rm d}t}f(t)\right)^{2}+4f(t)^{2}>0,\quad\mbox{and}$ (5.25)
$\displaystyle 2h(\theta)\tfrac{{\rm d}^{2}}{{\rm
d}\theta^{2}}h(\theta)-\left(\tfrac{{\rm d}}{{\rm
d}\theta}h(\theta)\right)^{2}+4h(\theta)^{2}>0;$ (5.26)

  2. (2)

$\theta(t)$ is a $D_{2d}$-equivariant orientation preserving diffeomorphism on
$\mathbb{R}/(2\mathbb{Z}\pi)$ which fixes each point in
$\tfrac{\mathbb{Z}\pi}{d}$;

  3. (3)

The triple $(f(t),h(\theta),\theta(t))$ is a solution of the following ODE
system for all $t\in\mathbb{R}/(2\mathbb{Z}\pi)$,

$\displaystyle\tfrac{1}{2f(t)}\tfrac{{\rm d}^{2}}{{\rm
d}t^{2}}f(t)-\tfrac{1}{4f(t)^{2}}\left(\tfrac{{\rm d}}{{\rm
d}t}f(t)\right)^{2}+1$ $\displaystyle=$ $\displaystyle\left(\tfrac{{\rm
d}}{{\rm d}t}\theta(t)\right)^{2}\ \left(\tfrac{1}{2h(\theta(t))}\tfrac{{\rm
d}^{2}}{{\rm
d}\theta^{2}}h(\theta(t))-\tfrac{1}{4h(\theta(t))^{2}}\left(\tfrac{{\rm
d}}{{\rm d}\theta}h(\theta(t))\right)^{2}+1\right),\quad\mbox{and}$
$\displaystyle\sin^{2}(t+\tfrac{k\pi}{d})+\tfrac{\cos(t+\tfrac{k\pi}{d})\sin(t+\tfrac{k\pi}{d})}{2f(t)}\tfrac{{\rm
d}}{{\rm d}t}f(t)$ $\displaystyle=$
$\displaystyle\sin^{2}(\theta(t)+\tfrac{k\pi}{d})+\tfrac{\cos(\theta(t)+\tfrac{k\pi}{d})\sin(\theta(t)+\tfrac{k\pi}{d})}{2h(\theta(t))}\tfrac{{\rm
d}}{{\rm d}\theta}h(\theta(t))$ (5.28)

for each $k\in\\{0,\cdots,d-1\\}$.

######  Remark 5.4.

The way we put Theorem 5.3 is explicit and convenient. However, it contains
some iterance. For example, when $d\in\\{1,2,3\\}$, we only need to keep the
ODE with $k=0$ for (5.28). When $d=2$, the two ODEs in (5.28) are equivalent,
because their sum is $1=1$. When $d=3$, we can use (5.28) with $k=0$ and the
$\mathbb{Z}_{d}$-symmetry in $D_{2d}$ to deduce the other ODEs in (5.26).
Similarly, when $d=4$ and $6$, we only need the two ODEs with $k=0,1$ for
(5.28).

###  5.3. Geometric description by (d)-property

In this subsection, we study the correspondence between the triple
$(F_{1},F_{2},\Phi)$ in Theorem 5.3 and its restriction to a normal plane
$\mathbf{V}$ for $M_{t}$. Recall that $\overline{F}_{1}=F_{1}|_{\mathbf{V}}$
and $\overline{F}_{2}=F_{2}|_{\mathbf{V}}$ are two $D_{2d}$-invariant
Minkowski norms on $\mathbf{V}$ with polar coordinates presentations
$\overline{F}_{1}=r\sqrt{2f(t)}$ and $\overline{F}_{2}=r\sqrt{2h(\theta)}$. In
Section 5.4, we have seen the restriction $\overline{\Phi}=\Phi|_{\mathbf{V}}$
is a Hessian isometry from $\overline{F}_{1}$ and $\overline{F}_{2}$ with the
polar coordinates presentation
$(r,t)\rightarrow(\tfrac{rf(t)^{1/2}}{h(\theta(t))^{1/2}},\theta(t))$. Here
$(f(t),h(\theta),\theta(t))$ is just the triple in Theorem 5.3 corresponding
to $(F_{1},F_{2},\Phi)$. The diffeomorphism $\theta(t)$ on
$\mathbb{R}/(2\mathbb{Z}\pi)$ is $D_{2d}$-equivariant and preserves the
orientation, so $\overline{\Phi}$ is $D_{2d}$-equivariant and preserves the
orientation as well. Further more, since $\theta(t)$ fixes each point in
$\tfrac{\mathbb{Z}\pi}{d}$, $\overline{\Phi}$ preserves each ray spanned by
the points in $\mathbf{V}\cap(M_{0}\cup M_{\pi/d})$.

The restriction to $\mathbf{V}$ provides the correspondence from the triple
$(F_{1},F_{2},\Phi)$ to the triple
$(\overline{F}_{1},\overline{F}_{2},\overline{\Phi})$, by which we can explain
(5.25)-(5.28) in Theorem 5.3.

Theorem 3.1 indicates the two inequalities (5.25) and (5.26) just tell us
$\overline{F}_{1}$ and $\overline{F}_{2}$ are Minkowski norms.

To explain (3), we denote
$\overline{T}_{1}=\partial_{t}-\tfrac{r}{2f(t)}\tfrac{{\rm d}}{{\rm
d}t}f(t)\partial_{r}$ and
$\overline{T}_{2}=\partial_{\theta}-\tfrac{r}{2h(\theta)}\tfrac{{\rm d}}{{\rm
d}\theta}h(\theta)\partial_{r}$ the tangent vector fields on
$\mathbf{V}\backslash\\{0\\}$ which generate $S_{\overline{F}_{1}}$ and
$S_{\overline{F}_{2}}$ respectively. Here $\partial_{r}$ and $\partial_{t}$
(or $\partial_{\theta}$ for $\overline{F}_{2}$) correspond to the polar $r$\-
and $t$-coordinates on $\mathbf{V}$. By Lemma 5.2 and the $D_{2d}$-symmetry,
we get $\overline{\Phi}_{*}(\overline{T}_{1})=\tfrac{{\rm d}}{{\rm
d}t}\theta(t)\ \overline{T}_{2}$. Then by (3.8) in Lemma 3.5, the ODE (3) just
tells us that
$g^{\overline{F}_{1}}_{x}(\overline{T}_{1},\overline{T}_{1})=g^{\overline{F}_{2}}_{\overline{\Phi}(x)}(\overline{\Phi}_{*}(\overline{T}_{1}),\overline{\Phi}_{*}(\overline{T}_{2}))$,
i.e., $\overline{\Phi}$ is a Hessian isometry between $\overline{F}_{i}$.

To explain (5.28), we recall that the orthonormal coordinates $(x_{1},x_{2})$
and polar coordinates $(r,t)$ (or $(r,\theta)$ where $\overline{F}_{2}$ is
concerned) of $x\in\mathbf{V}\backslash\\{0\\}$ are related by

$x=x_{1}v_{1}+x_{2}v_{2}=r\cos t\ v_{1}+r\sin t\ v_{2},$

in which $v_{1}\in\mathbf{V}\cap M_{0}$ and $v_{2}$ provide an orthonormal
basis on $\mathbf{V}$.

Denote $\overline{E}_{1}=\tfrac{1}{2}\overline{F}_{1}^{2}=r^{2}f(t)$, then we
have

$\displaystyle\tfrac{\partial}{\partial x_{1}}\overline{E}_{1}=\sin
t\tfrac{\partial}{\partial r}\overline{E}_{1}+\tfrac{1}{r}\cos
t\tfrac{\partial}{\partial t}\overline{E}_{1}=2r\sin tf(t)+r\cos t\tfrac{{\rm
d}}{{\rm d}t}f(t).$

So at $x=x_{1}v_{1}+x_{2}v_{2}\in S_{\overline{F}_{1}}$, where $2r^{2}f(t)=1$
and $x_{2}=r\sin t$,

$\displaystyle x_{2}\tfrac{\partial}{\partial x_{2}}\overline{E}_{1}=r\sin
t\left(2r\sin t\ f(t)+r\cos t\tfrac{{\rm d}}{{\rm
d}t}f(t)\right)=\sin^{2}t+\tfrac{\cos t\sin t}{2f(t)}\tfrac{{\rm d}}{{\rm
d}t}f(t),$

which coincides with the left side of (5.28). The right side of (5.28) can be
expressed similarly. So (5.28) tells us, with respect to the orthogonal
decomposition

$\mathbf{V}=\mathbf{V}^{\prime}+\mathbf{V}^{\prime\prime}=\mathbb{R}v_{1}+\mathbb{R}v_{2}\mbox{
(i.e., we have }\mathbf{V}^{\prime}=\mathbb{R}v_{1}\mbox{ and
}\mathbf{V}^{\prime\prime}=\mathbb{R}v_{2}\mbox{)},$

for any $x=x^{\prime}+x^{\prime\prime}\in S_{F_{1}}$ and
$\overline{\Phi}(x)=\overline{x}=\overline{x}^{\prime}+\overline{x}^{\prime\prime}\in
S_{F_{2}}$, the following equality is satisfied,

$g^{\overline{F}_{1}}_{x}(x^{\prime\prime},x)=g^{\overline{F}_{2}}_{\overline{x}}(\overline{x}^{\prime\prime},\overline{x}).$

More generally, we define this property as following.

######  Definition 5.5.

A Hessian isometry $\Phi$ between two Minkowski norms $F_{1}$ and $F_{2}$ on
$\mathbb{R}^{n}$ with $n\geq 2$ is said to satisfy the (d)-property with
respect to the orthogonal decomposition
$\mathbb{R}^{n}=\mathbf{V}^{\prime}+\mathbf{V}^{\prime\prime}$, if for any
nonzero $x=x^{\prime}+x^{\prime\prime}$ and
${\Phi}(x)=\overline{x}=\overline{x}^{\prime}+\overline{x}^{\prime\prime}$,
with $x^{\prime},\overline{x}^{\prime}\in\mathbf{V}^{\prime}$ and
$x^{\prime\prime},\overline{x}^{\prime\prime}\in\mathbf{V}^{\prime\prime}$, we
always have
$g^{{F}_{1}}_{x}(x^{\prime\prime},x)=g^{{F}_{2}}_{\overline{x}}(\overline{x}^{\prime\prime},\overline{x})$
(or equivalently,
$g^{{F}_{1}}_{x}(x^{\prime},x)=F_{1}(x)^{2}-g^{{F}_{1}}_{x}(x^{\prime\prime},x)=F_{2}(\overline{x})^{2}-g^{{F}_{2}}_{\overline{x}}(\overline{x}^{\prime\prime},\overline{x})=g^{{F}_{2}}_{\overline{x}}(\overline{x}^{\prime},\overline{x})$).

So the ODE (3) with $k=0$ can be interpreted as the (d)-property of
$\overline{\Phi}$ for
$\mathbf{V}=\mathbf{V}^{\prime}+\mathbf{V}^{\prime\prime}=\mathbb{R}v_{1}+\mathbb{R}v_{2}$.
By a similar argument, (5.28) with $0<k\leq d-1$ can be interpreted as the
(d)-property of $\overline{\Phi}$ for the decomposition

$\mathbf{V}=\mathbf{V}^{\prime}+\mathbf{V}^{\prime\prime}=\mathbb{R}(\cos(-\tfrac{k\pi}{d})v_{1}+\sin(-\tfrac{k\pi}{d})v_{2})+\mathbb{R}(\cos(\tfrac{\pi}{2}-\tfrac{k\pi}{d})v_{1}+\sin(\tfrac{\pi}{2}-\tfrac{k\pi}{d})v_{2}).$

To summarize, we see that $(F_{1},F_{2},\Phi)$ in Theorem 5.3 determines
$(\overline{F}_{1},\overline{F}_{2},\overline{\Phi})$ which satisfies
$D_{2d}$-symmetry and (d)-properties. Conversely, from any
$(\overline{F}_{1},\overline{F}_{2},\overline{\Phi})$ with $D_{2d}$-symmetry
and (d)-properties, we can retrieve the triple $(f(t),h(\theta),\theta(t))$ in
Theorem 5.3 and then use it to target $(F_{1},F_{2},\Phi)$.

Finally, we can transport
$(\overline{F}_{1},\overline{F}_{2},\overline{\Phi})$ to $\mathbb{R}^{2}$,
using the identification in Section 2.4, which identifies $v_{1}$ and $v_{2}$
to $e_{1}=(1,0)$ and $e=(0,1)$ in $\mathbb{R}^{2}$ respectively. By the
$D_{2d}$-symmetry, the triple after translation is irrelevant to the choice of
the identification between $\mathbf{V}$ and $\mathbb{R}^{2}$. It does not
depend on the choice of $\mathbf{V}$ either. So we have the one-to-one
correspondence in the following theorem.

######  Theorem 5.6.

Let $M_{t}$ be an isoparametric foliation on $(S^{n-1}(1),g^{\mathrm{st}})$
with $d$ principal curvatures. Then the restriction to a normal plane
$\mathbf{V}$ for $M_{t}$, and the identification between $\mathbf{V}$ and
$\mathbb{R}^{2}$ provide a one-to-one correspondence between the set of all
triples $(F_{1},F_{2},\Phi)$ satisfying the following:

  1. (1)

$F_{1}$ and $F_{2}$ are Minkowski norms on $\mathbb{R}^{n}$ induced by
$M_{t}$;

  2. (2)

$\Phi$ is a Hessian isometry from $F_{1}$ to $F_{2}$ which preserves the
orientation and fixes the spherical $\xi$-coordinates.

and the set of all triples
$(\overline{F}_{1},\overline{F}_{2},\overline{\Phi})$ satisfying the
following:

  1. (1)

$\overline{F}_{1}$ and $\overline{F}_{2}$ are two $D_{2d}$-invariant Minkowski
norms on $\mathbb{R}^{2}$;

  2. (2)

$\overline{\Phi}$ is a $D_{2d}$-equivariant orientation preserving Hessian
isometry from $\overline{F}_{1}$ to $\overline{F}_{2}$, which preserves each
ray spanned by $(\cos\tfrac{k\pi}{d},\sin\tfrac{k\pi}{d})$ for
$k\in\\{0,\cdots,2d-1\\}$.

  3. (3)

For each $k\in\\{0,\cdots,d-1\\}$, with respect to the decomposition

$\mathbb{R}^{2}=\mathbf{V}^{\prime}+\mathbf{V}^{\prime\prime}=\mathbb{R}(\cos(-\tfrac{k\pi}{d}),\sin(-\tfrac{k\pi}{d}))+\mathbb{R}(\cos(\tfrac{\pi}{2}-\tfrac{k\pi}{d},\sin(\tfrac{\pi}{2}-\tfrac{k\pi}{d}))),$

$\overline{\Phi}$ satisfies the (d)-property, i.e., for any nonzero
$x=x^{\prime}+x^{\prime\prime}$ and
$\overline{\Phi}(x)=\overline{x}=\overline{x}^{\prime}+\overline{x}^{\prime\prime}$,
with $x^{\prime},\overline{x}^{\prime}\in\mathbf{V}^{\prime}$ and
$x^{\prime\prime},\overline{x}^{\prime\prime}\in\mathbf{V}^{\prime\prime}$, we
always have
$g^{\overline{F}_{1}}_{x}(x^{\prime\prime},x)=g^{\overline{F}_{2}}_{\overline{x}}(\overline{x}^{\prime\prime},\overline{x})$.

######  Remark 5.7.

When $d=1$ or $2$, Theorem 5.6 only requires $\overline{\Phi}$ to satisfy the
(d)-property for the decomposition

$\mathbb{R}^{2}=\mathbf{V}^{\prime}+\mathbf{V}^{\prime\prime}=\mathbb{R}e_{1}+\mathbb{R}e_{2},$

i.e., when $d=2$, exchanging $\mathbf{V}^{\prime}$ and
$\mathbf{V}^{\prime\prime}$ does not count the second. On the other hand, when
$d>2$, essentially more (d)-properties for $\overline{\Phi}$ are required by
Theorem 5.6. This phenomenon and its consequence will be discussed in the next
two sections.

##  6\. Legendre transformation and (d)-property

###  6.1. Legendre transformation

In this paper, the Legendre transformation of a Minkowski norm $F$ on
$\mathbb{R}^{n}$ with $n\geq 2$ is referred to the following. By the strong
convexity of $F$, we have the following orientation preserving diffeomorphism,

$\Phi:\mathbb{R}^{n}\backslash\\{0\\}\rightarrow\mathbb{R}^{n}\backslash\\{0\\},\quad(x_{1},\cdots,x_{n})\mapsto(\tfrac{\partial}{\partial
x_{1}}E,\cdots,\tfrac{\partial}{\partial x_{n}}E),$ (6.29)

where $E=\tfrac{1}{2}F^{2}$. Obviously, $\Phi$ is (locally) linear if and only
if $F$ is (locally) Euclidean. The image $\Phi(S_{F})$ is a strongly convex
sphere surrounding the origin, so it determines a Minkowski norm $\hat{F}$ of
$F$ with $S_{\hat{F}}=\Phi(S_{F})$. We denote $g$ and $\hat{g}$ the Hessian
metrics of $F$ and $\hat{F}$ respectively. It is crucial to know that, with
respect to the coordinates
$x=(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}\backslash\\{0\\}$, the Hessian matrix
of $\hat{F}$ at $\overline{x}=\Phi(x)\in\mathbb{R}^{n}\backslash\\{0\\}$
coincides with the inverse matrix $(g^{ij})$ for the Hessian matrix $(g_{ij})$
of $F$ at $x$ (see Proposition 14.8.1 in [4] or Lemma 3.1.2 in [41]). This
observation, together with the fact
$\Phi_{*}({\partial_{x_{i}}})=\sum_{j}g_{ij}{\partial_{x_{j}}}$, implies

$\displaystyle\hat{g}(\Phi_{*}(\partial_{x_{i}}),\Phi_{*}(\partial_{x_{j}}))=\sum_{k,l}g_{ik}g_{jl}g^{kl}=g_{ij}=g_{1}(\partial_{x_{i}},\partial_{x_{j}}),$

i.e., $\Phi$ is a Hessian isometry [40]. On the other hand, $\Phi$ has the
following involutive property. If we denote
$\Phi(x)=\overline{x}=(\overline{x}_{1},\cdots,\overline{x}_{n})=(\sum_{i}g_{i1}x_{i},\cdots,\sum_{i}g_{in}x_{i})$
for $x=(x_{1},\cdots,x_{n})\in\mathbb{R}^{n}\backslash\\{0\\}$, then we see

$\displaystyle\Phi^{-1}(\overline{x}_{1},\cdots,\overline{x}_{n})=(x_{1},\cdots,x_{n})=(\sum_{i}g^{i1}\overline{x}_{i},\cdots,\sum_{i}g^{in}\overline{x}_{i})=(\tfrac{\partial}{\partial\overline{x}_{1}}\hat{E},\cdots,\tfrac{\partial}{\partial\overline{x}_{n}}\hat{E}),$

where $\hat{E}=\tfrac{1}{2}\hat{F}^{2}$, i.e., $\Phi^{-1}$ is the Legendre
transformation of $\hat{F}$.

To summarize, we call $\hat{F}$ the dual (Minkowski) norm of $F$ and $\Phi$
the Legendre transformation of $F$.

Notice that our notion in (6.29) has implicitly used the standard Euclidean
inner product to identify $\mathbb{R}^{n}$ with its dual. However, it does not
depend on the choice of orthonormal coordinates. So we have the following easy
lemma.

######  Lemma 6.1.

If a linear isometry on $(\mathbb{R}^{n},F)$ preserves the standard inner
product, then it commutes with the Legendre transformation of $F$ and
preserves the dual norm $\hat{F}$.

###  6.2. Hessian isometries satisfying all (d)-properties

From Theorem 5.6, we have seen the importance of the (d)-property. Indeed,
Legendre transformation is one of its origin.

######  Lemma 6.2.

Let $F_{1}$ and $F_{2}$ be two Minkowski norm on $\mathbb{R}^{n}$ with $n\geq
2$, and $\Phi$ a Hessian isometry from $F_{1}$ to $F_{2}$. If $\Phi$ is a
positive scalar multiplication or the composition between the Legendre
transformation of $F_{1}$ and a positive scalar multiplication, then $\Phi$
satisfies the (d)-property for every orthogonal decomposition.

Proof. Let $\mathbb{R}^{n}=\mathbf{V}^{\prime}+\mathbf{V}^{\prime\prime}$ be
any orthogonal decomposition. We prove the Hessian isometry in the lemma
satisfies the corresponding (d)-property.

Firstly, we prove the case $F_{2}(cx)=F_{1}(x)$ and $\Phi(x)=cx$ for some
constant $c>0$. With respect to above decomposition, if we have
$x=x^{\prime}+x^{\prime\prime}\in\mathbb{R}^{n}\backslash\\{0\\}$ then
$\overline{x}=\Phi(x)=cx=\overline{x}^{\prime}+\overline{x}^{\prime\prime}$
satisfies $\overline{x}^{\prime}=cx^{\prime}$ and
$\overline{x}^{\prime\prime}=cx^{\prime\prime}$. By the observation

$\displaystyle
g^{F_{2}}_{\overline{x}}(\overline{x}^{\prime\prime},\overline{x})$
$\displaystyle=$ $\displaystyle\tfrac{1}{2}\tfrac{{\rm d}}{{\rm
d}s}(F_{2}(\overline{x}+s\overline{x}^{\prime\prime})^{2})|_{s=0}=\tfrac{1}{2}\tfrac{{\rm
d}}{{\rm d}s}(F_{2}(cx+csx^{\prime\prime})^{2})|_{s=0}$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}\tfrac{{\rm d}}{{\rm
d}s}(F_{1}(x+sx^{\prime\prime})^{2})|_{s=0}=g^{F_{1}}_{x}(x^{\prime\prime},x).$

The (d)-property is proved.

Nextly, we prove the case that $\Phi$ is the Legendre transformation of
$F_{1}$. We may choose the orthonormal coordinates $(x_{1},\cdots,x_{n})$,
such that $\mathbf{V}^{\prime}$ and $\mathbf{V}^{\prime\prime}$ are given by
$x_{m+1}=\cdots=x_{n}=0$ and $x_{1}=\cdots=x_{m}=0$ respectively. Then for any
nonzero $x=(x_{1},\cdots,x_{n})=x^{\prime}+x^{\prime\prime}$ and
$\Phi(x)=\overline{x}=(\overline{x}_{1},\cdots,\overline{x}_{n})=\overline{x}^{\prime}+\overline{x}^{\prime\prime}$
with $x^{\prime},\overline{x}^{\prime}\in\mathbf{V}^{\prime}$ and
$x^{\prime\prime},\overline{x}^{\prime\prime}\in\mathbf{V}^{\prime\prime}$, we
have

$\displaystyle
g^{F_{2}}_{\overline{x}}(\overline{x}^{\prime\prime},\overline{x})$
$\displaystyle=$ $\displaystyle\sum_{m+1\leq i\leq n,1\leq j\leq
n}\left(\left(\sum_{k}x_{k}g_{ki}\right)\ g^{ij}\
\left(\sum_{l}x_{l}g_{jl}\right)\right)$ $\displaystyle=$
$\displaystyle\sum_{m+1\leq i\leq n,1\leq p\leq
n}x_{p}g_{pi}x_{i}=g^{F_{1}}_{x}(x^{\prime\prime},x),$

which proves the (d)-property of $\Phi$.

Finally, we prove the case that $\Phi$ is the composition between the Legendre
transformation of $F_{1}$ and a positive scalar multiplication. The argument
is a combination of above two, or one may apply Lemma 6.5 below.

Besides the Legendre transformation, identity maps, and their compositions
with positive scalar multiplications, there exists many other Hessian
isometries which satisfy the (d)-property for every orthogonal decomposition.
Here we propose a construction.

######  Example 6.3.

Let $C(U_{1})$ and $C(U_{2})$ be two conic open subsets in $\mathbb{R}^{n}$
with $n\geq 2$ such that their closures only intersect at the origin. We start
with the standard Euclidean norm $F_{0}$ on $\mathbb{R}_{n}$, and slightly
deform it on $C(U_{1})$ and $C(U_{2})$ to get the new Minkowski norm $F_{1}$.
The second new Minkowski norm $F_{2}$ is constructed by gluing $F_{1}$ on
$\mathbb{R}^{n}\backslash C(U_{1})$ and the dual norm of $F_{1}$ on
$C(U_{1})$. Then there is a Hessian isometry $\Phi$ from $F_{1}$ to $F_{2}$,
such that $\Phi$ coincides with the Legendre transformation of $F_{1}$ on
$C(U_{1})$ and the identity map elsewhere. This $\Phi$ satisfies the
(d)-property for every orthogonal decomposition. If $F_{1}$ is locally non-
Euclidean on $C(U_{1})$ and $C(U_{2})$, then $\Phi$ is not a positive scalar
multiplication or the composition between a Legendre transformation and a
positive scalar multiplication.

More examples can be constructed similarly, which may involve more (even
infinitely many) conic open subsets $C(U_{i})$ and different scalar changes.

###  6.3. Local (d)-property

For the convenience of later discussion, we also introduce the local version
for (d)-property.

######  Definition 6.4.

Let $F_{1}$ and $F_{2}$ be two Minkowski norms on $\mathbb{R}^{n}$ with $n\geq
2$, $C(U_{1})$ a connected conic open subset of
$\mathbb{R}^{n}\backslash\\{0\\}$, and
$\mathbb{R}^{n}=\mathbf{V}^{\prime}+\mathbf{V}^{\prime\prime}$ an orthogonal
decomposition. Then a local Hessian isometry $\Phi$ from $F_{1}$ to $F_{2}$ is
said to satisfy the local (d)-property on $C(U_{1})$ for
$\mathbb{R}^{n}=\mathbf{V}^{\prime}+\mathbf{V}^{\prime\prime}$, if $\Phi$ has
definition on $C(U_{1})$, and for any $x=x^{\prime}+x^{\prime\prime}\in
C(U_{1})$ and $\Phi(x)=\overline{x}=\overline{x}_{1}+\overline{x}_{2}$, with
$x^{\prime},\overline{x}^{\prime}\in\mathbf{V}^{\prime}$ and
$x^{\prime\prime},\overline{x}^{\prime\prime}\in\mathbf{V}^{\prime\prime}$, we
always have
$g^{\overline{F}_{1}}_{x}(x^{\prime\prime},x)=g^{\overline{F}_{2}}_{\overline{x}}(\overline{x}^{\prime\prime},\overline{x})$
(or equivalently,
$g^{\overline{F}_{1}}_{x}(x^{\prime},x)=F_{1}(x)^{2}-g^{\overline{F}_{1}}_{x}(x^{\prime\prime},x)=F_{2}(\overline{x})^{2}-g^{\overline{F}_{2}}_{\overline{x}}(\overline{x}^{\prime\prime},\overline{x})=g^{\overline{F}_{2}}_{\overline{x}}(\overline{x}^{\prime},\overline{x})$).

The following transitivity lemma for local (d)-property is easy to see.

######  Lemma 6.5.

Let $F_{i}$ with $1\leq i\leq 3$ be three Minkowski norms on $\mathbb{R}^{n}$
with $n\geq 2$, $C(U_{1})$ and $C(U_{2})$ two connected conic open subsets of
$\mathbb{R}^{n}\backslash\\{0\\}$, $\Phi_{1}$ is a local Hessian isometry from
$F_{1}$ to $F_{2}$, which maps $C(U_{1})$ into $C(U_{2})$, $\Phi_{2}$ is a
local Hessian isometry from $F_{2}$ to $F_{3}$ which has definition on
$C(U_{2})$. Then with respect to the same orthogonal decomposition
$\mathbb{R}^{n}=\mathbf{V}^{\prime}+\mathbf{V}^{\prime\prime}$,
$\Phi=\Phi_{2}\circ\Phi_{1}$ satisfies the (d)-property on $C(U_{1})$ when
each $\Phi_{i}$ satisfies the (d)-property on $C(U_{i})$.

We will need the following lemma in Section 7.2.

######  Lemma 6.6.

Let $F_{1}$ and $F_{2}$ be two Minkowski norms on $\mathbb{R}^{2}$, $C(U_{1})$
a connected conic open subset contained in the first quadrant
$\\{(x_{1},x_{2})|x_{1}>0,x_{2}>0\\}$, and $\Phi$ a local Hessian isometry
from $F_{1}$ to $F_{2}$. Suppose there exist positive constants $a$ and $b$,
such that the local Hessian isometry $\Phi$ from $F_{1}$ to $F_{2}$ can be
presented either as

$\Phi(x_{1},x_{2})=(ax_{1},bx_{2}),\quad\forall x=(x_{1},x_{2})\in C(U_{1})$

or as

$\Phi(x_{1},x_{2})=(a\tfrac{\partial}{\partial
x_{1}}E_{1},b\tfrac{\partial}{\partial x_{2}}E_{1}),\quad\forall
x=(x_{1},x_{2})\in C(U_{1}),$

where $E_{1}=\tfrac{1}{2}F_{1}^{2}$. If $\Phi$ satisfies the local
(d)-property on $C(U_{1})$ for the decomposition
$\mathbb{R}^{2}=\mathbf{V}^{\prime}+\mathbf{V}^{\prime\prime}$, in which
$\mathbf{V}^{\prime}$ and $\mathbf{V}^{\prime\prime}$ are spanned by $(\cos
c,\sin c)=\cos c\ e_{1}+\sin c\ e_{2}$ and $(-\sin c,\cos c)=-\sin c\
e_{1}+\cos c\ e_{2}$ respectively, for some $c\notin\mathbb{Z}\pi/2$, then
$\Phi|_{C(U_{1})}$ is either a positive scalar multiplication or the
composition between the Legendre transformation of $F_{1}$ and a positive
scalar multiplication.

Proof. We first prove Lemma 6.6 when $\Phi(x_{1},x_{2})=(ax_{1},bx_{2})$,
$\forall x=(x_{1},x_{2})\in C(U_{1})$.

Notice that this $\Phi$ satisfies the local (d)-property on $C(U_{1})$ for
$\mathbb{R}^{2}=\mathbf{V}^{\prime}+\mathbf{V}^{\prime\prime}=\mathbb{R}e_{1}+\mathbb{R}e_{2}$.
So we have

$\displaystyle
x_{2}g^{F_{1}}_{x}(e_{2},x)=\overline{x}_{2}g^{F_{2}}_{\overline{x}}(e_{2},\overline{x})\quad\mbox{and}\quad
x_{1}g^{F_{1}}_{x}(e_{1},x)=\overline{x}_{1}g^{F_{2}}_{\overline{x}}(e_{1},\overline{x})$
(6.30)

for any $x=(x_{1},x_{2})\in C(U_{1})$ and
$\Phi(x)=\overline{x}=(\overline{x}_{1},\overline{x}_{2})$.

On the other hand, the local (d)-property assumed in the lemma implies

$\displaystyle(-x_{1}\sin c+x_{2}\cos c)g_{x}^{F_{1}}(-\sin c\ e_{1}+\cos c\
e_{2},x)$ (6.31) $\displaystyle=$ $\displaystyle(-\overline{x}_{1}\sin
c+\overline{x}_{2}\cos c)g_{\overline{x}}^{F_{2}}(\sin c\ e_{1}+\cos c\
e_{2},\overline{x})$

for every

$x=(x_{1},x_{2})=(x_{1}\cos c+x_{2}\sin c)\ (\cos c,\sin c)+(-x_{1}\sin
c+x_{2}\cos c)\ (-\sin c,\cos c)\in C(U_{1})$

and

$\overline{x}=\Phi(x)=(\overline{x}_{1},\overline{x}_{2})=(\overline{x}_{1}\cos
c+\overline{x}_{2}\sin c)\ (\cos c,\sin c)+(-\overline{x}_{1}\sin
c+\overline{x}_{2}\cos c)\ (-\sin c,\cos c).$

Plug in (6.30), $\overline{x}_{1}=ax_{1}$ and $\overline{x}_{2}=bx_{2}$ into
(6.31), and use the fact that $\sin c\cos c\neq 0$ because
$c\notin\mathbb{Z}\pi/2$, we get

$a(a-b)x_{1}g_{x}^{F_{1}}(e_{2},x)=b(a-b)x_{2}g_{x}^{F_{1}}(e_{1},x).$

If $a=b$, there is nothing to prove. Otherwise, we get

$ax_{1}g_{x}^{F_{1}}(e_{2},x)=bx_{2}g_{x}^{F_{1}}(e_{1},x)$ (6.32)

for any $x=(x_{1},x_{2})\in C(U_{1})$.

Now we switch to the polar coordinates presentations $x=(x_{1},x_{2})=(r\cos
t,r\sin t)\in C(U_{1})$ and $F_{1}=r\sqrt{2f(t)}$, then (6.32) is translated
to the ODE

$(a\cos^{2}t+b\sin^{2}t)\tfrac{{\rm d}}{{\rm d}t}f(t)+(b-a)\sin 2tf(t)=0,$

which can be explicitly solved, i.e., for some positive constant $c^{\prime}$,
$f(t)=c^{\prime}\left|\tfrac{a+b}{a-b}+\cos 2t\right|$. Then we see
$F_{1}|_{C(U_{1})}$ coincides with the Euclidean norm
$\sqrt{\tfrac{4c^{\prime}}{|a-b|}\left(ax_{1}^{2}+bx_{2}^{2}\right)}$, i.e.,
$\Phi|_{C(U_{1})}$ coincides with the composition between the Legendre
transformation of $F_{1}$ and the scalar multiplication by
$\tfrac{|a-b|}{4c^{\prime}}$.

To summarize, we have proved Lemma 6.6 when
$\Phi(x_{1},x_{2})=(ax_{1},bx_{2})$, $\forall x=(x_{1},x_{2})\in C(U_{1})$. In
particular, we see $F_{1}|_{C(U_{1})}$ is a positive constant multiple of
$\sqrt{ax_{1}^{2}+bx_{2}^{2}}$ when $a\neq b$.

Then we prove Lemma 6.6 when $\Phi(x_{1},x_{2})=(a\tfrac{\partial}{\partial
x_{1}}E_{1},b\tfrac{\partial}{\partial x_{2}}E_{2})$ for any
$x=(x_{1},x_{2})\in C(U_{1})$.

Denote $C(U_{2})=\\{(x_{1},x_{2})|(ax_{1},bx_{2})\in C(U_{1})\\}$ another
connected conic open subset in the the first quadrant, $F_{3}$ the Minkowski
norm determined by $F_{3}(x,y)=F_{1}(ax,by)$, $\Phi_{1}(x,y)=(ax,by)$ the
linear isometry from $F_{3}$ to $F_{1}$, and $\Phi_{2}$ the Legendre
transformation of $F_{2}$. The composition $\Phi\circ\Phi_{1}$ coincides with
the Legendre transformation of $F_{3}$ on $C(U_{2})$. So by the involutive
property of Legendre transformation, $\Phi_{2}\circ\Phi$ maps $(x_{1},x_{2})$
to $(\tfrac{x_{1}}{a},\tfrac{x_{2}}{b})$ for any $x=(x_{1},x_{2})\in
C(U_{1})$. By Lemma 6.2 and Lemma 6.5, $\Phi_{2}\circ\Phi$ satisfies the local
(d)-property in Lemma 6.6. As we have proved above, either $a=b$, i.e.,
$\Phi|_{C(U_{1})}$ is the composition between the Legendre transformation of
$F_{1}$ and a positive scalar multiplication, or $F_{1}|_{C(U_{1})}$ is a
positive multiple of $\sqrt{\tfrac{x_{1}^{2}}{a}+\tfrac{x_{2}^{2}}{b}}$, i.e.,
$\Phi|_{C(U_{1})}$ is a positive scalar multiplication.

###  6.4. Legendre transformation of a Minkowski norm induced by $M_{t}$

Now we consider the Legendre transformation $\Phi$ of the Minkowski norm
$F_{1}=r\sqrt{2f(t)}$ induced by the isoparametric foliation $M_{t}$ on
$(S^{n-1}(1),g^{\mathrm{st}})$. We prove

######  Lemma 6.7.

Let $F_{1}=r\sqrt{2f(t)}$ be a Minkowski norm induced by the isoparametric
foliation $M_{t}$ on $(S^{n-1}(1),g^{\mathrm{st}})$ with $d$ principal
curvatures. Then for its dual norm $F_{2}$ and the Legendre transformation
$\Phi$ from $F_{1}$ to $F_{2}$, we have the following:

  1. (1)

$F_{2}$ is also a Minkowski norm induced by $M_{t}$;

  2. (2)

$\Phi$ preserves the orientation and fixes the spherical $\xi$-coordinates;

  3. (3)

Let $\mathbf{V}$ be any normal plane for $M_{t}$, then the restriction
$\overline{\Phi}=\Phi|_{\mathbf{V}}$ coincides with the Legendre
transformation of $\overline{F}_{1}=F_{1}|_{\mathbf{V}}$.

Proof. With the normal plane $\mathbf{V}$ for $M_{t}$ arbitrarily chosen, we
can find the orthonormal coordinates $(x_{1},\cdots,x_{n})$ for
$\mathbb{R}^{n}$ such that $\mathbf{V}$ is given by $x_{3}=\cdots=x_{n}=0$ and
$(1,0,\cdots,0)\in M_{0}$. Around any point
$x\in\mathbf{V}\backslash(\mathbb{R}_{\geq 0}M_{0}\cup\mathbb{R}_{\geq
0}M_{\pi/d})$, we can define a spherical local frame induced by $M_{t}$. The
values of $X_{1},\cdots,X_{n-2}$ span the same tangent subspace in
$T_{x}(\mathbb{R}^{n}\backslash\\{0\\})$ as those of
$\partial_{x_{3}},\cdots,\partial_{x_{n}}$. By Lemma 3.4 and continuity, we
have

$\tfrac{\partial}{\partial
x_{i}}E_{1}|_{\mathbf{V}\backslash\\{0\\}}=0,\quad\forall 3\leq i\leq n,$

where $E_{1}=\tfrac{1}{2}F_{1}^{2}$. So we have
$\Phi(x_{1},x_{2},0,\cdots,0)=(\tfrac{\partial}{\partial
x_{1}}E_{1},\tfrac{\partial}{\partial x_{2}}E_{1},0,\cdots,0)$, from which we
see that $\Phi$ preserves $\mathbf{V}\backslash\\{0\\}$ and
$\overline{\Phi}=\Phi|_{\mathbf{V}\backslash\\{0\\}}$ is the Legendre
transformation for $\overline{F}_{1}=F_{1}|_{\mathbf{V}_{1}}$.

For different normal planes for $M_{t}$, $\overline{F}_{1}$ corresponds to the
same Minkowski norm on $\mathbb{R}^{2}$, and then, so its dual norm
$\overline{F}_{2}=F_{2}|_{\mathbf{V}_{1}}$ and its Legendre transformation
$\overline{\Phi}=\Phi|_{\mathbf{V}}$ are irrelevant to the choice of
$\mathbf{V}$. This observation implies $F_{2}$ is a Minkowski norm induced by
$M_{t}$, and $\Phi$ preserves the foliation.

Since $\overline{F}_{1}$ is $D_{2d}$-invariant, its Legendre transformation
$\overline{\Phi}$, is $D_{2d}$-equivariant and orientation preserving, and it
preserves the ray spanned by each point in
$\mathbf{V}\cap(S^{n-1}(1)\backslash(M_{0}\cup M_{\pi/d}))$. Let
$(r,t)\mapsto(\tfrac{rf(t)^{1/2}}{h(\theta)^{1/2}},\theta(t))$ be the polar
coordinates presentation for $\overline{\Phi}$. Then $\theta$ fixes each point
in $\tfrac{\mathbb{Z}\pi}{d}$ and preserves the interval
$(\tfrac{k\pi}{d},\tfrac{(k+1)\pi}{d})$ for each $k\in\\{0,\cdots,2d-1\\}$.
Each interval $(\tfrac{k\pi}{d},\tfrac{(k+1)\pi}{d})$ for $t$ determines a
conic open subset of $\mathbf{V}\backslash\\{0\\}$ which corresponds to a
distinct spherical $\xi$-coordinate in $\mathbb{R}^{n}$. With $\mathbf{V}$
changing arbitrarily, we see from this observation that $\Phi$ fixes the
spherical $\xi$-coordinates. Finally, from the spherical coordinates
presentation
$(r,t,\xi)\mapsto(\tfrac{rf(t)^{1/2}}{h(\theta(t))^{1/2}},\theta(t),\xi)$ for
$\Phi$, which contains the same $\theta(t)$ but restrict $t$ to
$(0,\tfrac{\pi}{d})$, we see $\Phi$ preserves the orientation.

Lemma 6.7 indicates that the correspondence in Theorem 5.6 relates Legendre
transformations to Legendre transformations. On the other hand, for any
Legendre transformation $\overline{\Phi}$ between the $D_{2d}$-invariant
Minkowski norms $\overline{F}_{1}$ on $\mathbb{R}^{2}$ to its dual
$\overline{F}_{2}$, the (d)-property requirement in Theorem 5.6 is met by
Lemma 6.1 and Lemma 6.2. So $\overline{\Phi}$ is related to a Hessian isometry
$\Phi$ from $F_{1}$ to $F_{2}$. It must be the same one as in Lemma 6.7, i.e.,
the Legendre transformation of $F_{1}$, because the two coincide when
restricted to each normal plane for $M_{t}$.

To summarize, we have the following one-to-one correspondence between Legendre
transformations, which is not affected by the choice for the normal plane
$\mathbf{V}$ for $M_{t}$ or its identification with $\mathbb{R}^{2}$.

######  Theorem 6.8.

Let $M_{t}$ be an isoparametric foliation on $(S^{n-1}(1),g^{\mathrm{st}})$
with $d$ principal curvatures. Then the restriction to any normal plane
$\mathbf{V}$ for $M_{t}$ and the identification between $\mathbf{V}$ and
$\mathbb{R}^{2}$ provide a one-to-one correspondence between the set of
Legendre transformations of Minkowski norms on $\mathbb{R}^{n}$ induced by
$M_{t}$, and the set of Legendre transformations of $D_{2d}$-invariant
Minkowski norms on $\mathbb{R}^{2}$.

######  Remark 6.9.

The function $\theta(t)$ in the polar coordinates presentation
$(r,t)\mapsto(\tfrac{rf(t)^{1/2}}{h(\theta)^{1/2}},\theta(t))$ for the
Legendre transformation
$\overline{\Phi}(x_{1},x_{2})=(\tfrac{\partial}{\partial
x_{1}}E_{1},\tfrac{\partial}{\partial x_{2}}E_{1})$ for a $D_{2d}$-invariant
$F_{1}=r\sqrt{2f(t)}$, with $E_{1}=\tfrac{1}{2}F_{1}^{2}=r^{2}f(t)$, has been
calculated (with more generality) in [56], i.e.,

$\theta(t)=\arccos\left(\tfrac{2\cos tf(t)-\sin t\tfrac{{\rm d}}{{\rm
d}t}f(t)}{\left(4f(t)^{2}+\left(\tfrac{{\rm d}}{{\rm
d}t}f(t)\right)^{2}\right)^{1/2}}\right),$ (6.33)

when $t\in(0,\pi)$. It is a solution of

$\tfrac{{\rm d}}{{\rm d}t}\theta(t)=\tfrac{\left(2f(t)\tfrac{{\rm d}^{2}}{{\rm
d}t^{2}}f(t)-\left(\tfrac{{\rm d}}{{\rm
d}t}f(t)\right)^{2}+4f(t)^{2}\right)\sin\theta(t)\cos\theta(t)}{\left(\cos
t\tfrac{{\rm d}}{{\rm d}t}f(t)+2\sin tf(t)\right)\left(-\sin t\tfrac{{\rm
d}}{{\rm d}t}f(t)+2\cos tf(t)\right)},$ (6.34)

for $t\in(0,\tfrac{\pi}{d})$. The general solutions of (6.34) have also been
given by [56], i.e.,

$\displaystyle\theta(t)=\arccos\left(\tfrac{2\cos tf(t)-\sin t\tfrac{{\rm
d}}{{\rm
d}t}f(t)}{\left[4(\cos^{2}t+\tfrac{b^{2}}{a^{2}}\sin^{2}t)f(t)^{2}+4(\tfrac{b^{2}}{a^{2}}-1)\cos
t\sin tf(t)\tfrac{{\rm d}}{{\rm
d}t}f(t)+(\sin^{2}t+\tfrac{b^{2}}{a^{2}}\cos^{2}t)\left(\tfrac{{\rm d}}{{\rm
d}t}f(t)\right)^{2}\right]^{1/2}}\right),$ (6.35)

which appears in the polar coordinates presentation for the mapping

$(x_{1},x_{2})\mapsto(a\tfrac{\partial}{\partial
x_{1}}E_{1},b\tfrac{\partial}{\partial x_{1}}E_{1}),$

with positive constants $a$ and $b$.

##  7\. ODE method and local description

###  7.1. ODE method

In this section, we analysis the ODE system for the triple
$(f(t),h(\theta),\theta(t))$ consisting of (3) and (5.28) in Theorem 5.3,
i.e.,

$\displaystyle\tfrac{1}{2f(t)}\tfrac{{\rm d}^{2}}{{\rm
d}t^{2}}f(t)-\tfrac{1}{4f(t)^{2}}\left(\tfrac{{\rm d}}{{\rm
d}t}f(t)\right)^{2}+1$ $\displaystyle=$ $\displaystyle\left(\tfrac{{\rm
d}}{{\rm d}t}\theta(t)\right)^{2}\ \left(\tfrac{1}{2h(\theta(t))}\tfrac{{\rm
d}^{2}}{{\rm
d}\theta^{2}}h(\theta(t))-\tfrac{1}{4h(\theta(t))^{2}}\left(\tfrac{{\rm
d}}{{\rm d}\theta}f(t)\right)^{2}+1\right),$
$\displaystyle\sin^{2}(t+\tfrac{k\pi}{d})+\tfrac{\cos(t+\tfrac{k\pi}{d})\sin(t+\tfrac{k\pi}{d})}{2f(t)}\tfrac{{\rm
d}}{{\rm d}t}f(t)$ $\displaystyle=$
$\displaystyle\sin^{2}(\theta(t)+\tfrac{k\pi}{d})+\tfrac{\cos(\theta(t)+\tfrac{k\pi}{d})\sin(\theta(t)+\tfrac{k\pi}{d})}{2h(\theta(t))}\tfrac{{\rm
d}}{{\rm d}\theta}h(\theta(t)),\ \forall k\in\\{0,\cdots,d-1\\},$ (7.37)

where $d$ is number of principal curvatures for $M_{t}$. Recall that
$(f(t),h(\theta),\theta(t))$ is from the spherical coordinates presentations
for the triple $(F_{1},F_{2},\Phi)$ in Theorem 5.3, and it determines the
triple $(\overline{F}_{1},\overline{F}_{2},\overline{\Phi})$ in Theorem 5.6.

While discussing this ODE system, we assume $f(t)$ satisfying the requirements
in Theorem 3.1 has been arbitrarily chosen and discuss how to locally
determine $\theta(t)$ and $h(\theta)$. To avoid some minor technical
complexity, we fix some $t_{0}\in(0,\tfrac{\pi}{d})$ (so we have
$\theta_{0}=\theta(t_{0})\in(0,\tfrac{\pi}{d})$ as well), and discuss the ODE
system for $t$ in a sufficiently small neighborhood of
$t_{0}\in(0,\tfrac{\pi}{d})$. We assume $d>2$ because the cases $d=1$ and
$d=2$ have been discussed by Theorem 1.4 and Theorem 1.5 in [56]. The ODE
method in [56] for discussing (7.1) and (7.37) with $k=0$ is still a main
ingredient here. We sketch it as following.

Rewrite (7.37) with $k=0$ as

$\displaystyle\tfrac{1}{h(\theta(t))}\tfrac{{\rm d}}{{\rm
d}\theta}h(\theta(t))=\left({2\sin^{2}t+\tfrac{\cos t\sin t}{f(t)}\tfrac{{\rm
d}}{{\rm d}t}f(t)}\right){\csc\theta(t)\sec\theta(t)}-2\tan\theta(t),$ (7.38)

and differentiate it with respect to $t$, then we get

$\displaystyle\tfrac{{\rm d}}{{\rm d}t}\theta(t)\
\left(\tfrac{1}{h(\theta(t))}\tfrac{{\rm d}^{2}}{{\rm
d}\theta^{2}}h(\theta(t))-\tfrac{1}{h(\theta(t))^{2}}\left(\tfrac{{\rm
d}}{{\rm d}\theta}h(\theta(t))\right)^{2}\right)$ $\displaystyle=$
$\displaystyle\tfrac{{\rm d}}{{\rm d}t}\theta(t)\
{\left(2\sin^{2}t+\tfrac{\cos t\sin t}{f(t)}\tfrac{{\rm d}}{{\rm
d}t}f(t)\right)\left(\sec^{2}\theta(t)-\csc^{2}\theta(t)\right)}-2\tfrac{{\rm
d}}{{\rm d}t}\theta(t)\ \sec^{2}\theta(t)$ $\displaystyle+\left({4\cos t\sin
t+\tfrac{\cos^{2}t-\sin^{2}t}{f(t)}\tfrac{{\rm d}}{{\rm d}t}f(t)-\tfrac{\cos
t\sin t}{f(t)^{2}}\left(\tfrac{{\rm d}}{{\rm d}t}f(t)\right)^{2}+\tfrac{\cos
t\sin t}{f(t)}\tfrac{{\rm d}^{2}}{{\rm
d}t^{2}}f(t)}\right){\csc\theta(t)\sec\theta(t)}.$

We plug (7.38) and (7.1) into the right side of (7.1), to erase $h(\theta(t))$
and its derivatives, then we get a formal quadratic equation for $\tfrac{{\rm
d}}{{\rm d}t}\theta(t)$,

$A\left(\tfrac{{\rm d}}{{\rm d}t}\theta(t)\right)^{2}+B\left(\tfrac{{\rm
d}}{{\rm d}t}\theta(t)\right)+C=0,$ (7.40)

in which

$\displaystyle A$ $\displaystyle=$ $\displaystyle\frac{\cos t\sin t\left(\cos
t\tfrac{{\rm d}}{{\rm d}t}f(t)+2\sin tf(t)\right)\left(\sin t\tfrac{{\rm
d}}{{\rm d}t}f(t)-2\cos
tf(t)\right)}{2f(t)^{2}\cos^{2}\theta(t)\sin^{2}\theta(t)},$ $\displaystyle B$
$\displaystyle=$ $\displaystyle\frac{\tfrac{\cos t\sin t}{f(t)}\tfrac{{\rm
d}^{2}}{{\rm d}t^{2}}f(t)-\tfrac{\cos t\sin t}{f(t)^{2}}\left(\tfrac{{\rm
d}}{{\rm d}t}f(t)\right)^{2}+\tfrac{\cos^{2}t-\sin^{2}t}{f(t)}\tfrac{{\rm
d}}{{\rm d}t}f(t)+4\cos t\sin t}{\cos\theta(t)\sin\theta(t)},$ $\displaystyle
C$ $\displaystyle=$ $\displaystyle-\tfrac{1}{f(t)}\tfrac{{\rm d}^{2}}{{\rm
d}t^{2}}f(t)+\tfrac{1}{2f(t)^{2}}\left(\tfrac{{\rm d}}{{\rm
d}t}f(t)\right)^{2}-2.$

Since we have assumed $d>2$, for $t\in(0,\tfrac{\pi}{d})$, we have
$\theta(t)\in(0,\tfrac{\pi}{d})\subset(0,\tfrac{\pi}{2})$. So from (6.34) in
Remark 6.9, we see that $\left(\cos t\tfrac{{\rm d}}{{\rm d}t}f(t)+2\sin
tf(t)\right)\left(\sin t\tfrac{{\rm d}}{{\rm d}t}f(t)-2\cos tf(t)\right)$ is a
positive factor in $A$ for $t\in(0,\tfrac{\pi}{d})$. To summarize, when $d>2$,
the $A$-coefficient in (7.40) is always nonzero for each
$t\in(0,\tfrac{\pi}{d})$.

Direct calculation shows that for each $t$ close to $t_{0}$, the two solutions
of (7.40) are

$\frac{\cos\theta(t)\sin\theta(t)}{\cos t\sin
t}\quad\mbox{and}\quad\tfrac{\left(-2f(t)\tfrac{{\rm{\rm d}^{2}}}{{\rm
d}t^{2}}f(t)+\left(\tfrac{{\rm d}}{{\rm
d}t}f(t)\right)^{2}-4f(t)^{2}\right)\cos\theta(t)\sin\theta(t)}{\left(\cos
t\tfrac{{\rm d}}{{\rm d}t}f(t)+2\sin tf(t)\right)\left(\sin t\tfrac{{\rm
d}}{{\rm d}t}f(t)-2\cos tf(t)\right)}.$ (7.41)

The discriminant of (7.40) is

$B^{2}-4AC=\left(\frac{\cos t\sin t\tfrac{{\rm d}^{2}}{{\rm
d}t^{2}}f(t)+(\sin^{2}t-\cos^{2}t)\tfrac{{\rm d}^{2}}{{\rm
d}t}f(t)}{\cos\theta(t)\sin\theta(t)}\right)^{2},$

which vanishes if and only if $\cos t\sin t\tfrac{{\rm d}^{2}}{{\rm
d}t^{2}}f(t)+(\sin^{2}t-\cos^{2}t)\tfrac{{\rm d}^{2}}{{\rm d}t}f(t)=0$.

###  7.2. Case-by-case discussion

There are two generic possibilities.

Case 1. We have $\cos t_{0}\sin t_{0}\tfrac{{\rm d}^{2}}{{\rm
d}t^{2}}f(t)+(\sin^{2}t_{0}-\cos^{2}t_{0})\tfrac{{\rm d}^{2}}{{\rm
d}t}f(t)\neq 0$. Then we have (see Lemma 3.7 in [56])

######  Lemma 7.1.

If $\cos t_{0}\sin t_{0}\tfrac{{\rm d}^{2}}{{\rm
d}t^{2}}f(t)+(\sin^{2}t_{0}-\cos^{2}t_{0})\tfrac{{\rm d}^{2}}{{\rm
d}t}f(t)\neq 0$, then one of the following two cases must happen:

  1. (1)

For all $t$ sufficiently close to $t_{0}$, we have

$\tfrac{{\rm d}}{{\rm d}t}\theta(t)=\tfrac{\cos\theta(t)\sin\theta(t)}{\cos
t\sin t};$ (7.42)

  2. (2)

For all $t$ sufficiently close to $t_{0}$, we have

$\tfrac{{\rm d}}{{\rm d}t}\theta(t)=\tfrac{\left(-2f(t)\tfrac{{\rm{\rm
d}^{2}}}{{\rm d}t^{2}}f(t)+\left(\tfrac{{\rm d}}{{\rm
d}t}f(t)\right)^{2}-4f(t)^{2}\right)\cos\theta(t)\sin\theta(t)}{\left(\cos
t\tfrac{{\rm d}}{{\rm d}t}f(t)+2\sin tf(t)\right)\left(\sin t\tfrac{{\rm
d}}{{\rm d}t}f(t)-2\cos tf(t)\right)}.$ (7.43)

Each of (7.42) and (7.43) has a unique solution for the initial value problem
$\theta_{0}=\theta(t_{0})\in(0,\tfrac{\pi}{d})$. Then input this solution
$\theta(t)$ into (7.37) with $k=0$, and use the initial value condition
$h(\theta_{0})=h_{0}>0$, we can locally determine $h(\theta)$ around
$\theta_{0}$, as well as the Hessian isometry
$\overline{\Phi}:(r,t)\mapsto(\tfrac{rf(t)^{1/2}}{h(\theta(t))^{1/2}},\theta(t))$
from $\overline{F}_{1}=r\sqrt{2f(t)}$ to $\overline{F}_{2}=r\sqrt{2h(\theta)}$
on $\mathbb{R}^{2}$.

Subcase 1.1. (7.42) is satisfied for all $t$ sufficiently close to $t_{0}$.

The ODE (7.42) has the solution

$\theta(t)=\arccos\left(\tfrac{a\cos
t}{(a^{2}\cos^{2}t+b^{2}\sin^{2}t)^{1/2}}\right)$

with suitable positive constants $a$ and $b$ to meet all the initial value
requirements. In this case, $\overline{\Phi}$ coincides with the linear
isomorphism $(x_{1},x_{2})\mapsto(ax_{1},bx_{2})$ when polar the
$t$-coordinate is close to $t_{0}$.

Here comes the speciality of $d>2$. By Theorem 5.6, $\overline{\Phi}$
satisfies the (d)-property for the orthonormal decomposition
$\mathbb{R}^{2}=\mathbf{V}^{\prime}+\mathbf{V}^{\prime\prime}=\mathbb{R}(\cos(-\tfrac{\pi}{d}),\sin(-\tfrac{\pi}{d}))+\mathbb{R}(\cos(\tfrac{\pi}{2}-\tfrac{\pi}{d}),\sin(\tfrac{\pi}{2}-\tfrac{\pi}{d}))$.
So by Lemma 6.6, $\overline{\Phi}$ locally coincides with a positive scalar
multiplication or the composition between the Legendre transformation of
$\overline{F}_{1}$ and a positive scalar multiplication.

Subcase 1.2. (7.43) is satisfied for all $t$ sufficiently close to $t_{0}$.

The solution $\theta(t)$ for (7.43) is provided in (6.35), in which the
positive parameters $a$ and $b$ can be suitably chosen to meet all initial
value requirements. The corresponding $\overline{\Phi}$ is given by
$\overline{\Phi}(x_{1},x_{2})=(a\tfrac{\partial}{\partial
x_{1}}E_{1},b\tfrac{\partial}{\partial x_{1}}E_{1})$ with
$E_{1}=\tfrac{1}{2}F_{1}^{2}$ when the polar $t$-coordinate of
$x=(x_{1},x_{2})$ is sufficiently close to $t_{0}$. Using Theorem 5.6 and
Lemma 6.6 for the speciality of $d>2$ again, we see that $\overline{\Phi}$
locally coincides with a positive scalar multiplication or the composition