a

Iterations of Minkowski Valuations

Oscar Ortega-Moreno

>
>
>
> Abstract. It is shown that for any sufficiently regular even Minkowski
> valuation $\Phi$ which is homogeneous and intertwines rigid motions, and for
> any convex body $K$ in a smooth neighborhood of the unit ball, there exists
> a sequence of positive numbers $(\gamma_{m})_{m=1}^{\infty}$ such that
> $(\gamma_{m}\Phi^{m}K)_{m=1}^{\infty}$ converges to the unit ball with
> respect to the Hausdorff metric.

1\. Introduction

In the theory of convex bodies, a central role is played by the so-called
projection body operator. To states its precise definition, let us recall that
a convex body $K$ (that is, a compact, convex set with non-empty interior) in
$\mathbb{R}^{n}$ is uniquely determined by its support function
$h_{K}(u)=\max\\{u\cdot x:x\in K\\}$, $u\in\mathbb{S}^{n-1}$. For a given
convex body $K$, the support function of the projection body of $K$, denoted
by $\Pi K$, is defined by $h_{\Pi
K}(x)=\textrm{vol}_{n-1}\left(K|x^{\perp}\right)$ for all
$x\in\mathbb{S}^{n-1}$, where $K|x^{\perp}$ is the projection of $K$ onto the
hyperplane with normal unit vector $x$. The projection body was introduced by
Minkowski himself and was later discovered to be of great importance in a
number of different areas (see, e.g.,[12], [26], and [42]).

Petty’s conjectured volume inequality for projection bodies [37] remains a
major open problem in convex and integral geometry. It states that the
quotient $V_{n}(\Pi K)/V_{n}(K)^{n-1}$ is minimized when $K$ is an ellipsoid.
A positive answer to this conjecture would not only lead to a new
isoperimetric inequality for convex sets, but also to an inequality stronger
than a number of old and new isoperimetric inequalities, including important
inequalities such as the polar projection inequality of Petty and the affine
isoperimetric inequality.

The class reduction technique, first introduced by Schneider in [41], reduces
Petty’s conjectured inequality to the study of a fixed point problem. More
precisely, this approach consists in finding solutions to the equation
$\Pi^{2}K=\alpha K$ where $\alpha>0$. This condition characterizes possible
minimizers of the quotients $V_{n}(\Pi K)/V_{n}(K)^{n-1}$ and narrows the
search for such minimizers down to the class of zonoids, which is the range of
the projection body operator.

As Lutwak observed in [32, 31], the class reduction technique can be
generalized to projection bodies of different degrees. For
$i\in\\{1,\dots,n-1\\}$ and a given convex body $K$, the support function of
the projection body of $K$ of order $i$, denoted by $\Pi_{i}K$, is defined by
$h_{\Pi_{i}K}(x)=V_{i}\left(K|x^{\perp}\right)$ for all
$x\in\mathbb{S}^{n-1}$, where $V_{i}$ denotes the $i$-th intrinsic volume. For
$i\in\\{1,\dots,n-1\\}$, Lutwak [32] conjectured that the quotient
$V_{i+1}(\Pi_{i}K)/V_{i+1}(K)^{i}$ is minimized among all convex bodies, when
$K$ is a Euclidean ball. In fact, he showed that these chain of inequalities
would follow from Petty’s conjectured inequality. Furthermore, he extended the
class reduction technique to characterize minimizers for these quotients by
proving that a minimizer must also satisfy the fixed point equation
$\Pi_{i}^{2}K=\alpha K$ for some $\alpha>0$. In conjunction with a result by
Schneider [40], who showed that the only solutions to $\Pi_{1}^{2}K=\alpha K$
are balls, Lutwak [32] confirmed his conjecture for the case $i=1$ by showing
that the quotient $V_{2}(\Pi_{1}K)/V_{2}(K)$ is bounded from below.

While the solution to the equation $\Pi_{1}^{2}K=\alpha K$ are only balls, it
comes as a surprised that the projection body operator $\Pi=\Pi_{n-1}$ admits
a variety of smooth and non-smooth solutions, with balls and cubes being the
most basic examples. A complete characterization of polytopal solutions to the
fixed point problem $\Pi^{2}K=\alpha K$ was provided by Weil [50], who showed
that they are the orthogonal sums of symmetric polygons and segments.
Surprisingly, there is not much known about the smooth solutions. Recently,
Ivaki [22, 23] made a major step towards this direction by showing that
locally around the unit ball, the only smooth solution to the problem are
ellipsoids. This also follows from a stronger result by Saroglou and Zvavitch
[38], who considered limits of the sequence of iterations of the projection
body operator with respect to the Banach-Mazur distance. With their result
they manage to confirm Petty’s conjectured inequality locally around the unit
ball for smooth bodies. The main goal of this article is to extend the results
of Saroglou and Zvavich [38] on the convergence of the iterations of the
projection body operator to a larger class of Minkowski valuations. A
_Minkowski valuation_ on the space $\mathcal{K}^{n}$ of convex bodies in
$\mathbb{R}^{n}$ endowed with the Hausdorff metric is a map
$\Phi:\mathcal{K}^{n}\to\mathcal{K}^{n}$ such that

$\Phi K+\Phi L=\Phi(K\cup L)+\Phi(K\cap L)$

whenever $K\cup L$ is convex and addition is the usual Minkowski addition.

The most celebrated theorem in valuation theory is undoubtedly Hadwiger’s
classic theorem [20] on the characterization of continuous, $\mathrm{SO}(n)$
invariant, real valued valuations as linear combinations of intrinsic volumes.
It provided the kick-start needed to the systematic study of valuations and
laid the groundwork for a number of venues of research in convex, discrete,
and integral geometry (see, e.g., [2, 3, 4, 7, 17, 25, 29])

The investigation of Minkowski valuations is of a rather recent vintage. It
started in 1974 with a paper by Schneider [39] on Minkowski endomorphisms:
Minkowski additive valuations which commute with rigid motions. Nonetheless,
it was the groundbreaking work by Ludwig [27, 28] that prompted a wave of
further developments. In her paper [27], Ludwig solved a question posed by
Lutwak on the characterization of the projection body operator as the unique
continuous, translation invariant and affinely contravariant Minkowski
valuation.

Following Ludwig’s steps, many authors have confirmed a basic principle when
it comes to characterizing Minkowski valuations compatible with volume
preserving linear maps: they often form a convex cone generated by finitely
many transformations (see, e.g., [1, 8, 9, 16, 30, 45, 49]). As opposed to
affine compatiblitity, the less restrictive condition of being merely rotation
equivariant means that the characterization of Minkowski valuation with such
property will encompass a larger class of transformations: a cone with
infinitely many generators. This feature makes the problem of finding an
analogue of Hadwiger’s theorem for Minkowski valuations a challenging one.
Several authors have made substantial progress in this direction (see [24, 43,
44, 46, 47]) by showing that Minkowski valuations admit a certain _spherical
convolution_ representation (see Section 2). More recently, Dorrek [10]
established the following remarkable theorem under the additional assumption
of homogeneity. Throughout, a map
$\Phi:\mathcal{K}^{n}\rightarrow\mathcal{K}^{n}$ is said to have degree $i$ if
$\Phi(\lambda K)=\lambda^{i}\Phi(K)$ for all $K\in\mathcal{K}$ and
$\lambda\geq 0$. (By a result of McMullen, any translation invariant
continuous valuation that is also homogeneous must be of integer degree
$i\in\\{0,\ldots,n\\}$.)

###### Theorem 1.1

([10]) If $\Phi_{i}:\mathcal{K}^{n}\rightarrow\mathcal{K}^{n}$ is a
continuous translation invariant Minkowski valuation of degree $1\leq i\leq
n-1$ which commutes with $\mathrm{SO}(n)$, then there exists a unique
$\mathrm{SO}(n-1)$ invariant $f\in L^{1}(\mathbb{S}^{n-1})$ with center of
mass at the origin such that for every $K\in\mathcal{K}^{n}$,

$h_{\Phi_{i}K}=S_{i}(K,\cdot)\ast f.$ (1.1)

Here $S_{i}(K,\cdot)$, $1\leq i\leq n-1$, is the _area measure of order $i$_
on $\mathbb{S}^{n-1}$ associated to $K$ (see Section 2). We call the function
$f$ in (1.1) the _generating function_ of $\Phi_{i}$. A complete
classification of all such generating functions is still an important open
problem in valuation theory. However, it is known that the support function of
an arbitrary convex body of revolution $L\in\mathcal{K}^{n}$ is the generating
function of a Minkowski valuation of degree $i$. In this case, we say that
$\Phi_{i}$ is generated by $L$. If, in addition, for some $m\geq 2$, the
boundary of $L$ is a $C^{m}$ submanifold of $\mathbb{R}^{n}$ with everywhere
positive Gaussian curvature, we call $\Phi_{i}$ a _$C^{m}_{+}$ regular_
Minkowski valuation. If $\Phi_{i}K=\\{o\\}$ for all $K\in\mathcal{K}^{n}$, we
call $\Phi_{i}$ _trivial_.

In recent years, several classic inequalities involving projection bodies of
arbitrary degree have been shown to hold for large (if not all) subclasses of
Minkowski valuations intertwining rigid motions (see, e.g., [5, 6, 19, 21, 36,
43]). Some of these results are indeed a consequence of already known
inequalities for the projection bodies, which turn out to be the limiting
cases of such families of inequalities. In a recent article[35], the approach
suggested by the author together with Schuster is the opposite one: showing
that the inequality holds for a large and well behaved family of Minkowski
valuations with which one can approximate the projection body operator.

Conjecture 1 ([35]) Let
_$\Phi_{i}:\mathcal{K}^{n}\rightarrow\mathcal{K}^{n}$ be a continuous
translation invariant Minkowski valuation of degree $1\leq i\leq n-1$ which
commutes with $\mathrm{SO}(n)$. Among convex bodies $K\subseteq\mathbb{R}^{n}$
of non-empty interior, the volume ratio $V_{i+1}(\Phi_{i}K)/V_{i+1}(K)^{i}$ is
minimized when $K$ is a Euclidean ball._

As pointed out in [35], this conjecture provides a new approach to Petty’s
conjecture. First of all, notice that the projection body map can by
approximated by $C^{\infty}_{+}$ regular Minkowski valuations. Thus, if the
conjecture is proven for a sufficiently large class of these smooth
valuations, we can obtain Petty’s conjectured inequality (up to equality
cases) by taking a limit. There are a couple of reasons why this smoothing
approach seems advantageous. On one hand, if $\Phi_{i}$ is a smooth valuation,
then for any convex body $K$, its image $\Phi K$ belongs to the same class of
smoothness. This eliminates all possible polytopal solutions of the fixed
point problem and thus reduces the set of possible minimizers of the quotient.
On the other hand, the additional regularity of the operators allows us to use
analytic machinery to study the problem (indeed, as we will see later, this is
already reflected in the local solution around the unit ball).

A further generalization of the Schneider and Lutwak class reduction technique
to Minkowski valuations was obtain in [35]:

Proposition 1 ([35]) _Let $1\leq i\leq n-1$ and
$\Phi_{i}:\mathcal{K}^{n}\rightarrow\mathcal{K}^{n}$ be a non-trivial monotone
and translation invariant Minkowski valuation of degree $i$ which commutes
with $\mathrm{SO}(n)$. If $K\in\mathcal{K}^{n}$ has dimension at least $i+1$,
then_

$\frac{V_{i+1}(\Phi_{i}K)}{V_{i+1}(K)^{i}}\geq\frac{V_{i+1}(\Phi_{i}^{2}K)}{V_{i+1}(\Phi_{i}K)^{i}}$
(1.2)

_with equality if and only if $\Phi_{i}^{2}K$ and $K$ are homothetic.
Moreover, if $\Phi_{i}$ is $C_{2}^{+}$ regular and balls are the only
solutions to the fixed-point problem $\Phi_{i}^{2}K=\alpha K$ for some
$\alpha>0$, then $V_{i+1}(\Phi_{i}K)/V_{i+1}(K)^{i}$ is minimized precisely by
Euclidean balls._

The problem to determine the convergence of iterations of Minkowski valuations
is itself an intriguing one. Nevertheless, its importance becomes
unquestionable once one realizes its connection to Conjecture 1 via the class
reduction technique (Proposition 1). To explain this in more detail, let us
first state the main result of this article.

###### Theorem 1

Let $2\leq i\leq n-1$ and $\Phi_{i}:\mathcal{K}^{n}\to\mathcal{K}^{n}$ be a
$C^{\infty}_{+}$ regular translation invariant even Minkowski valuation of
degree $i$ which commutes with $\mathrm{SO}(n)$. Then, there exists
$\varepsilon>0$ such that if $K\in\mathcal{K}^{n}$ has support function of
class $C^{2}$ and satisfies $\|h_{\gamma
K}-h_{\mathbb{B}^{n}}\|_{C^{2}(\mathbb{S}^{n-1})}<\varepsilon$ for some
$\gamma>0$, then there exists a sequence of positive numbers
$(\gamma_{m})_{m=1}^{\infty}$ such that

$d_{H}(\gamma_{m}\Phi_{i}^{m}K,\mathbb{B}^{n})\to 0\quad\text{as
}\,\,m\to\infty.$

Here $d_{H}(K,L)$ denotes the Hausdorff distance of two given sets $K$ and
$L$. As a simple consequence of Theorem 1 we obtain a local solution to the
fixed point problem $\Phi^{2}K=\alpha K$ for smooth Minkowski valuations. This
provides a different proof to the main results of [35]. However, in [35] the
fixed point problem in question is solved for a bigger class of Minkowski
valuations that also includes valuations that are not necessarily smooth.

Corollary 1 _Let $2\leq i\leq n-1$ and
$\Phi_{i}:\mathcal{K}^{n}\to\mathcal{K}^{n}$ be a $C^{\infty}_{+}$ regular
translation invariant even Minkowski valuation of degree $i$ which commutes
with $\mathrm{SO}(n)$. Then there exists $\varepsilon>0$ such that if
$K\in\mathcal{K}^{n}$ has $C^{2}$ support function and satisfies $\|h_{\gamma
K}-h_{\mathbb{B}^{n}}\|_{C^{2}(\mathbb{S}^{n-1})}<\varepsilon$ for some
$\gamma>0$, and_

$\Phi_{i}^{2}K=\alpha K,$

_for some $\alpha>0$, then $K$ is a ball. _

Furthermore, Theorem 1 leads to a local solution to Conjecture 1 using the
class reduction technique. Note that Corollary 1 by itself does not suffice to
obtain a local solution of the conjecture, as it only characterizes possible
minimizers locally: balls can still be a different kind of critical points for
the quotient. The proof relies heavily on the local convergence of the
iterations.

Corollary 2 _Let $2\leq i\leq n-1$ and
$\Phi_{i}:\mathcal{K}^{n}\to\mathcal{K}^{n}$be a $C^{\infty}_{+}$ regular
translation invariant even Minkowski valuation of degree $i$ which commutes
with $\mathrm{SO}(n)$. Then there exists $\varepsilon>0$ such that if
$K\in\mathcal{K}^{n}$ has $C^{2}$ support function and satisfies $\|h_{\gamma
K}-h_{\mathbb{B}^{n}}\|_{C^{2}(\mathbb{S}^{n-1})}<\varepsilon$ for some
$\gamma>0$, then_

$\frac{V_{i+1}(\Phi_{i}K)}{V_{i+1}(K)^{i}}\geq\frac{V_{i+1}(\Phi_{i}\mathbb{B}^{n})}{V_{i+1}(\mathbb{B}^{n})^{i}},$

_with equality if and only if $K$ is a ball. _

The reader might be wondering why the case $i=1$ is excluded from the
statement of the previous theorems. This is due to the fact that, in this
case, the convergence holds globally on $\mathcal{K}^{n}$.

###### Theorem 2

Let $\Phi_{1}:\mathcal{K}^{n}\to\mathcal{K}^{n}$ be a translation invariant
monotone Minkowski valuation of degree $1$ which commutes with $SO(n)$ with
generating function $g$ of class $C^{2}$. Then, for all $K\in\mathcal{K}^{n}$

$d_{H}\left(\gamma_{m}\Phi_{1}^{m}K,\mathbb{B}^{n}\right)\to 0\quad\text{as
}\,\,m\to\infty,$

with
$\gamma_{m}^{-1}=\frac{w(K)}{2}\left(\int_{\mathbb{S}^{n-1}}g(u)du\right)^{m}$,
where $w(K)$ denotes the mean width of $K$.

The paper is organized as follows. In Section 2, we recall some basic
background material from convex geometry, harmonic analysis and approximation
theory. In Section 3, we state and prove some auxiliary results that will be
needed for the proof of our main results. Finally, in the last section we
prove the main theorem of this article as well as its corollaries.

2\. Background material

The aim of this section is to recall some terminology and notation as well as
standard results from convex geometry and harmonic analysis. Excellent and
comprehensive presentations of the material cited here are the monographs by
Gardner [12], Schneider [42], and Groemer [15], with the latter focusing
specifically on harmonic analysis and its applications to convex geometry.

Convex bodies and Support functions. A convex body is a convex compact subset
of $\mathbb{R}^{n}$ with non-empty interior. We denote by $\mathcal{K}^{n}$
the set of all convex bodies in $\mathbb{R}^{n}$. Each convex body
$K\in\mathcal{K}^{n}$ is determined by its support function

$h_{K}(x)=\max\\{x\cdot y:y\in K\\},\quad x\in\mathbb{R}^{n},$

which is (positively) homogeneous of degree one and subadditive. Conversely,
every function on $\mathbb{R}^{n}$ with these properties is the support
function of a unique convex body in $\mathcal{K}^{n}$. In particular, a
homogeneous function $h\in C^{2}(\mathbb{R}^{n})$ of degree one is the support
function of a convex body $K\in\mathcal{K}^{n}$ if and only if its Hessian
$\nabla^{2}h(u)$ is positive semi-definite for all $u\in\mathbb{S}^{n-1}$.
Note that the gradient of a $1$-homogeneous function is a $0$-homogeneous map
and so $\nabla^{2}f(u)u=0$ for all $u\in\mathbb{S}^{n-1}$. The Hessian of $f$
is therefore determined by the restriction of the map $\nabla^{2}f(u)$ to
$u^{\perp}$. We will always refer to this restriction as $D^{2}f(u)$ for all
$u\in\mathbb{S}^{n-1}$.

The Minkowski sum of $K,L\in\mathcal{K}^{n}$ is defined by

$K+L=\\{x+y:x\in K,y\in L\\}.$

It is easy to check that the support function of $K+L$ is given by
$h_{K+L}=h_{K}+h_{L}.$ For every $\vartheta\in\mathrm{SO}(n)$ and
$y\in\mathbb{R}^{n}$, we have

$h_{\vartheta K}(x)=h_{K}(\vartheta^{-1}x)\qquad\mbox{and}\qquad
h_{K+y}=h_{K}(x)+x\cdot y$

for all $x\in\mathbb{R}^{n}$. Moreover, $K\subseteq L$ if and only if
$h_{K}\leq h_{L}$, in particular, $h_{K}>0$ if and only if
$o\in\mathrm{int}\,K$.

Higher regularity. A body $K\in\mathcal{K}^{n}$ is said to be of class
$C^{k}_{+}$ if its boundary hypersurface $\partial K$ is a $C^{k}$ submanifold
of $\mathbb{R}^{n}$ and the map $n_{K}:\partial K\to\mathbb{S}^{n-1}$ that
maps a boundary point to its unique outer unit normal is a $C^{k}$
diffeomorphism. Equivalently, $K\in\mathcal{K}^{n}$ is of class $C^{k}_{+}$ if
$h_{K}\in C^{k}(\mathbb{R}^{n})$ and the restriction of the Hessian
$\nabla^{2}h_{K}(u)$ to $u^{\perp}$ is positive definite for every
$u\in\mathbb{S}^{n-1}$ i.e, $D^{2}h_{K}(u)$ is positive definite for every
$u\in\mathbb{S}^{n-1}$. We mention here that for any smooth function
$f:\mathbb{S}^{n-1}\to\mathbb{R}$, we denote by $\nabla^{2}f(u)$ the Hessian
of its 1 homogeneous extension and $D^{2}f(u)$ the restriction of
$\nabla^{2}f(u)$ to $u^{\perp}$. Moreover, if $\nabla^{2}_{\mathbb{S}}f$
denotes the spherical Hessian, then $D^{2}f=\nabla^{2}_{\mathbb{S}}f+fI$,
where $I$ denotes the identity.

Mixed volumes and area measures. One of the pillars of the Brunn–Minkowski
theory of convex bodies is the fact that the volume of a Minkowski linear
combination $\lambda_{1}K_{1}+\cdots+\lambda_{m}K_{m}$, where
$K_{1},\ldots,K_{m}\in\mathcal{K}^{n}$ and $\lambda_{1},\ldots,\lambda_{m}\geq
0$, can be expressed as a homogeneous polynomial of degree $n$,

$V_{n}(\lambda_{1}K_{1}+\cdots+\lambda_{m}K_{m})=\sum\limits_{j_{1},\ldots,j_{n}=1}^{m}V(K_{j_{1}},\ldots,K_{j_{n}})\lambda_{j_{1}}\cdots\lambda_{j_{n}},$

where the coefficients $V(K_{j_{1}},\ldots,K_{j_{n}})$ are the mixed volumes
of $K_{j_{1}},\ldots,K_{j_{n}}$, which depend only on
$K_{j_{1}},\ldots,K_{j_{n}}$ and are symmetric in their arguments. Moreover,
mixed volumes are translation invariant, Minkowski additive, monotone w.r.t.
set inclusion in each of their arguments, and $V(K_{1},\ldots,K_{n})>0$ if and
only if there are segments $l_{i}\subseteq K_{i}$, $1\leq i\leq n$, with
linearly independent directions.

For $K,L\in\mathcal{K}^{n}$ and $0\leq i\leq n$, let $V(K[i],L[n-i])$ denote
the mixed volume with $i$ copies of $K$ and $n-i$ copies of $L$. The _$i$ th
intrinsic volume_ of $K$ is given by

$V_{i}(K)=\frac{1}{\kappa_{n-i}}\binom{n}{i}V(K[i],\mathbb{B}^{n}[n-i]),$

where $\kappa_{m}$ denotes the $m$-dimensional volume of $\mathbb{B}^{m}$.

Associated with an $(n-1)$-tuple of bodies
$K_{2},\ldots,K_{n}\in\mathcal{K}^{n}$ is a finite Borel measure
$S(K_{2},\ldots,K_{n},\cdot)$ on $\mathbb{S}^{n-1}$, the mixed area measure,
such that for all $K_{1}\in\mathcal{K}^{n}\\!$,

$V(K_{1},\ldots,K_{n})=\frac{1}{n}\int\displaylimits_{\mathbb{S}^{n-1}}h(K_{1},u)\,dS(K_{2},\ldots,K_{n},u).$
(2.1)

For $K\in\mathcal{K}^{n}$ and $0\leq i\leq n-1$, the measures
$S_{i}(K,\cdot):=S(K[i],\mathbb{B}^{n}[n-1-i],\cdot)$ are called the _area
measures_ of order $i$ of $K$. The measure $S_{n-1}(K,\cdot)$ is also known as
the _surface area measure_ of $K$. If $K$ has non-empty interior, then, by a
theorem of Aleksandrov–Fenchel–Jessen (see, e.g., [42, p. 449]), each of the
measures $S_{i}(K,\cdot)$, $1\leq i\leq n-1$, determines $K$ up to
translations. The centroid of every area measure of a convex body is at the
origin, that is, for every $K\in\mathcal{K}^{n}$ and all
$i\in\\{0,\ldots,n-1\\}$,

$\int\displaylimits_{\mathbb{S}^{n-1}}u\,dS_{i}(K,u)=o.$

_Minkowski’s existence theorem_ states that a non-negative Borel measure $\mu$
on $\mathbb{S}^{n-1}$ is the surface area measure of some
$K\in\mathcal{K}^{n}$ with non-empty interior if and only if $\mu$ is not
concentrated on a great subsphere of $\mathbb{S}^{n-1}$ and has centroid at
the origin.

Area densities and mixed discriminants. If $K\in\mathcal{K}^{n}$ has a $C^{2}$
support function, then each measure $S_{i}(K,\cdot)$, $0\leq i\leq n-1$, is
absolutely continuous w.r.t. spherical Lebesgue measure. To make this more
precise, let us recall the notion of mixed discriminants. If
$A_{1},\ldots,A_{m}$ are symmetric real $k\times k$ matrices and
$\lambda_{1},\dots,\lambda_{m}\geq 0$, then

$\det(\lambda_{1}A_{1}+\cdots+\lambda_{m}A_{m})=\sum_{j_{1},\dots,j_{k}=1}^{m}\mathrm{D}(A_{j_{1}},\ldots,A_{j_{k}})\lambda_{j_{1}}\cdots\lambda_{j_{k}},$
(2.2)

where the coefficients $\mathrm{D}(A_{j_{1}},\ldots,A_{j_{k}})$ are the mixed
discriminants of $A_{j_{1}},\ldots,A_{j_{k}}$ which depend only on
$A_{j_{1}},\ldots,A_{j_{k}}$ and are symmetric and multilinear in their
arguments. Clearly, $\mathrm{D}(A,\dots,A)=\det(A)$ for any symmetric $k\times
k$ matrix $A$. Moreover,
$\mathrm{D}(BA_{1},\ldots,BA_{k})=\det(B)\mathrm{D}(A_{1},\ldots,A_{k})$,

$\mathrm{D}(A,B,\ldots,B)=\frac{1}{n-1}\mathrm{tr}(\mathrm{cof}(B)A)$ (2.3)

for any symmetric $k\times k$ matrix $B$, and if $A_{1},\ldots,A_{k}$ are
positive semi-definite, then $\mathrm{D}(A_{1},\dots,A_{k})\geq 0$. Finally,
if $K_{1},\ldots,K_{n-1}\in\mathcal{K}^{n}$ have support functions
$h_{1},\ldots,h_{n-1}\in C^{2}(\mathbb{R}^{n})$, then the density of
$S(K_{1},\ldots,K_{n-1},\cdot)$ is given by

$s(K_{1},\ldots,K_{n-1},u)=\mathrm{D}(D^{2}h_{1}(u),\ldots,D^{2}h_{n-1}(u)),\quad
u\in\mathbb{S}^{n-1}.$ (2.4)

In particular, for $K\in\mathcal{K}^{n}$ with support function $h\in
C^{2}(\mathbb{R}^{n})$, we have

$s_{n-1}(K,u)=\det D^{2}h(u),\quad u\in\mathbb{S}^{n-1}.$ (2.5)

Motivated by (2.4) and (2.5), we frequently use in subsequent sections the
notation $s(h_{1},\ldots,h_{n-1},\cdot)$, $s_{n-1}(h,\cdot)$, $\ldots$ instead
of $s(K_{1},\ldots,K_{n-1},\cdot)$, $s_{n-1}(K,u)$, etc.

Metrics on the space $\mathcal{K}^{n}$. The most used notion of convergence on
convex bodies is the one derived from the Hausdorff metric. Given any pair of
non-empty sets $X,Y\subset\mathbb{R}^{n}$, their Hausdorff distance is defined
by

$d_{H}(X,Y)=\inf\\{\varepsilon\geq 0\,;\ X\subseteq Y_{\varepsilon}\text{ and
}Y\subseteq X_{\varepsilon}\\},$

where

$X_{\varepsilon}=\bigcup_{x\in X}\\{z\in M\,;\ \|z-x\|_{2}\leq\varepsilon\\}.$

It is well known that the Hausdorff metric of two bodies
$K,L\in\mathcal{K}^{n}$ can be expressed as

$d_{H}(K,L)=\|h_{K}-h_{L}\|_{\infty},$

where $\|\,\cdot\,\|_{\infty}$ denotes the maximum norm on
$C(\mathbb{S}^{n-1})$.

Naturally, a further family of metrics is derived by replacing the maximum
norm by an $L^{p}$ norm. For $p\in[1,\infty)$ and $K,L\in\mathcal{K}^{n}$, let

$d_{p}(K,L)=\left(\int_{\mathbb{S}^{n-1}}|h_{K}(u)-h_{L}(u)|^{p}du\right)^{\frac{1}{p}}.$

Clearly, $d_{p}$ is a metric on $\mathcal{K}^{n}$, called the $L_{p}$ metric.
The $L_{2}$ metric is of particular interest in this context because of the
well-developed Harmonic analysis on $\mathbb{S}^{n-1}$. In general, $L_{p}$
metrics do not necessarily induce the same notion of convergence; however,
with the additional structure provided by the support functions, Vitale [48]
deduced that all of the $d_{p}$ metrics, $p\in[1,\infty]$, generate the same
topology on $\mathcal{K}^{n}$ and yield a complete metric spaces in which
closed, bounded sets are compact (see Theorem 3 in [48]).

There are other distances on convex bodies that emerge from distances between
measures. By the Aleksandrov–Fenchel–Jessen uniqueness theorem (see, e.g.,
[42, p. 449]), a convex body is uniquely determined, up to translation, by its
area measure of order $i$ for all $i\in\\{1,\dots,n-1\\}$. Therefore, any
metric in the space of positive measures $\mathcal{M}(\mathbb{S}^{n-1})$ gives
rise to a metric on the set of convex bodies with centroid at the origin. Some
examples of such metrics, that will play an important role in the sequel are
the total variation and the Lévy–Prokhorov metric. The total variation metric
between to positive measures $\mu,\nu\in\mathcal{M}(\mathbb{S}^{n-1})$ is
given by

$d_{\mathrm{TV}}(\mu,\nu)=\sup_{A\in\mathcal{B}(\mathbb{S}^{n-1})}\left|\mu(A)-\nu(A)\right|$

where $\mathcal{B}(\mathbb{S}^{n-1})$ denotes the set of Borel sets on
$\mathbb{S}^{n-1}$. The Lévy–Prokhorov metric is a weaker metric on the space
of positive measures on the sphere $\mathcal{M}_{+}(\mathbb{S}^{n-1})$. It is
defined by

$d_{\mathrm{LP}}(\mu,\nu)=\inf\left\\{\varepsilon>0:\mu(A)\leq\nu(A_{\varepsilon})+\varepsilon\text{
and
}\nu(A)\leq\mu(A_{\varepsilon})+\varepsilon,A\in\mathcal{B}(\mathbb{S}^{n-1})\right\\},$

where $A_{\varepsilon}=\\{u\in\mathbb{S}^{n-1}:d(A,u)<\varepsilon\\}$. The
metric $d_{\mathrm{LP}}$ has the property that
$d_{\mathrm{LP}}(\mu_{k},\mu)\to 0$ if and only if $(\mu_{k})_{k=1}^{\infty}$
converges weakly to $\mu$. It is easy to see that the convergence with respect
to $d_{TV}$ implies convergence with respect to $d_{LP}$.

Harmonic analysis on $\mathbb{S}^{n-1}$. We turn now to the background
material on spherical harmonics. To this end, let $\Delta_{\mathbb{S}}$ denote
the spherical Laplacian on $\mathbb{S}^{n-1}$ and recall that it is a second-
order uniformly elliptic self-adjoint operator. We write $\mathcal{H}_{k}^{n}$
for the vector space of spherical harmonics of dimension $n$ and degree $k$
and denote its dimension by

$N(n,k)=\frac{n+2k-2}{n+k-2}{n+k-2\choose n-2}=\mathrm{O}(k^{n-2})\mbox{ as
}k\rightarrow\infty.$ (2.6)

Spherical harmonics are (precisely) the eigenfunctions of
$\Delta_{\mathbb{S}}$, more specific, for $Y_{k}\in\mathcal{H}_{k}^{n}$, we
have

$\Delta_{\mathbb{S}}Y_{k}=-k(k+n-2)\,Y_{k}.$ (2.7)

The spaces $\mathcal{H}_{k}^{n}$ are pairwise orthogonal subspaces of
$L^{2}(\mathbb{S}^{n-1})$. Moreover, the Fourier series
$f\sim\sum_{k=0}^{\infty}\pi_{k}f$ converges to $f$ in $L^{2}$ for every $f\in
L^{2}(\mathbb{S}^{n-1})$, where
$\pi_{k}:L^{2}(\mathbb{S}^{n-1})\rightarrow\mathcal{H}_{k}^{n}$ denotes the
orthogonal projection. Letting $P_{k}^{n}\in C([-1,1])$ denote the _Legendre
polynomial_ of dimension $n$ and degree $k$, we have

$(\pi_{k}f)(v)=\frac{N(n,k)}{\omega_{n}}\int\displaylimits_{\mathbb{S}^{n-1}}\\!\\!f(u)\,P_{k}^{n}(u\cdot
v)\,du,\qquad v\in\mathbb{S}^{n-1},$ (2.8)

where $\omega_{n}$ denotes the surface area of $\mathbb{B}^{n}$ and
integration is with respect to spherical Lebesgue measure. Since the
orthogonal projection $\pi_{k}$ is self adjoint, it is consistent to extend it
to the space $\mathcal{M}(\mathbb{S}^{n-1})$ of signed finite Borel measures
by

$(\pi_{k}\mu)(v)=\frac{N(n,k)}{\omega_{n}}\int\displaylimits_{\mathbb{S}^{n-1}}\\!\\!P^{n}_{k}(u\cdot
v)\,d\mu(u),\qquad v\in\mathbb{S}^{n-1}.$

It can be shown easily that $\pi_{k}\mu\in\mathcal{H}_{k}^{n}$ for all $k\geq
0$ and that the formal Fourier series $\mu\sim\sum_{k=0}^{\infty}\pi_{k}\mu$
uniquely determines the measure $\mu$.

Throughout, we use $\bar{e}\in\mathbb{S}^{n-1}$ to denote a fixed but
arbitrarily chosen pole of $\mathbb{S}^{n-1}$ and write $\mathrm{SO}(n-1)$ for
the stabilizer in $\mathrm{SO}(n)$ of $\bar{e}$. In the theory of spherical
harmonics, a function or measure on $\mathbb{S}^{n-1}$ which is
$\mathrm{SO}(n-1)$ invariant is often called _zonal_. Clearly, zonal functions
depend only on the value of $u\cdot\bar{e}$. The subspace of zonal functions
in $\mathcal{H}_{k}^{n}$ is $1$-dimensional for every $k\geq 0$ and spanned by
$u\mapsto P_{k}^{n}(u\cdot\bar{e})$. Since the spaces $\mathcal{H}_{k}^{n}$
are orthogonal, it is not difficult to show that any zonal measure
$\mu\in\mathcal{M}(\mathbb{S}^{n-1})$ admits a series expansion of the form

$\mu\sim\sum_{k=0}^{\infty}\frac{N(n,k)}{\omega_{n}}\,a_{k}^{n}[\mu]\,P_{k}^{n}(\,\,.\cdot\bar{e}),$
(2.9)

where

$a_{k}^{n}[\mu]=\omega_{n-1}\int\displaylimits_{-1}^{1}P_{k}^{n}(t)\,(1-t^{2})^{\frac{n-3}{2}}\,d\bar{\mu}(t).$
(2.10)

Here, we have used cylindrical coordinates $u=t\bar{e}+\sqrt{1-t^{2}}v$ on
$\mathbb{S}^{n-1}$ to identify the zonal measure $\mu$ with a measure
$\bar{\mu}$ on $[-1,1]$. If $\mu$ is absolutely continuous with density $f$
w.r.t. spherical Lebesgue measure, we write $a_{k}^{n}[f\,]$ instead of
$a_{k}^{n}[\mu]$.

For the explicit computation of integrals of the form (2.10) the following
_Formula of Rodrigues_ for the Legendre polynomials is often useful:

$P_{k}^{n}(t)=(-1)^{k}\frac{\Gamma\left(\frac{n-1}{2}\right)}{2^{k}\Gamma\left(\frac{n-1}{2}+k\right)}(1-t^{2})^{-\frac{n-3}{2}}\left(\frac{d}{dt}\right)^{k}(1-t^{2})^{\frac{n-3}{2}+k}.$
(2.11)

Multiplier and integral transforms. The well known _Funk–Hecke Theorem_ states
that if $f\in C([-1,1])$ and
$\mathrm{T}_{f}:\mathcal{M}(\mathbb{S}^{n-1})\rightarrow C(\mathbb{S}^{n-1})$
is defined by

$(\mathrm{T}_{f}\mu)(u)=\int\displaylimits_{\mathbb{S}^{n-1}}\\!\\!f(u\cdot
v)\,d\mu(v),\qquad u\in\mathbb{S}^{n-1},$ (2.12)

then the spherical harmonic expansion of $\mathrm{T}_{f}\mu$ is given by

$\mathrm{T}_{f}\mu\sim\sum_{k=0}^{\infty}a_{k}^{n}[f]\,\pi_{k}\mu,$ (2.13)

where the numbers $a_{k}^{n}[f]$ are given by (2.10) and called the
_multipliers_ of $\mathrm{T}_{f}$.

Integral transforms of the form (2.12) are closely related to the convolution
between functions and measures on $\mathbb{S}^{n-1}$. In order to recall its
definition, first note that the convolution $\sigma\ast\tau$ of signed
measures $\sigma,\tau$ on the compact Lie group $\mathrm{SO}(n)$ can be
defined by

$\int\displaylimits_{\mathrm{SO}(n)}\\!\\!\\!f(\vartheta)\,d(\sigma\ast\tau)(\vartheta)=\int\displaylimits_{\mathrm{SO}(n)}\\!\int\displaylimits_{\mathrm{SO}(n)}\\!\\!\\!f(\eta\theta)\,d\sigma(\eta)\,d\tau(\theta),\qquad
f\in C(\mathrm{SO}(n)).$

By identifying $\mathbb{S}^{n-1}$ with the homogeneous space
$\mathrm{SO}(n)/\mathrm{SO}(n-1)$, one obtains a one-to-one correspondence of
$C(\mathbb{S}^{n-1})$ and $\mathcal{M}(\mathbb{S}^{n-1})$ with right
$\mathrm{SO}(n-1)$ invariant functions and measures on $\mathrm{SO}(n)$,
respectively. Using this correspondence, the convolution of measures on
$\mathrm{SO}(n)$ induces a convolution product on $\mathcal{M}(S^{n-1})$ (for
more details see, e.g., [43]). For this spherical convolution, zonal functions
and measures play a particularly important role. Let us therefore denote by
$C(\mathbb{S}^{n-1},\bar{e})$ the set of continuous zonal functions on
$\mathbb{S}^{n-1}$. Then, for $\mu\in\mathcal{M}(\mathbb{S}^{n-1})$, $f\in
C(\mathbb{S}^{n-1},\bar{e})$, and $\eta\in\mathrm{SO}(n)$, we have

$(\mu\ast
f)(\eta\bar{e})=\int\displaylimits_{\mathbb{S}^{n-1}}\\!\\!f(\eta^{-1}u)\,d\mu(u).$
(2.14)

Note that, by (2.14), we have $(\vartheta\mu)\ast f=\vartheta(\mu\ast f)$ for
every $\vartheta\in\mathrm{SO}(n)$, where $\vartheta\mu$ is the image measure
of $\mu$ under $\vartheta\in\mathrm{SO}(n)$. Moreover, from the identification
of a zonal function $f$ on $\mathbb{S}^{n-1}$ with a function $\bar{f}$ on
$[-1,1]$, (2.10), and (2.14), we obtain

$a_{k}^{n}[f]=\int\displaylimits_{\mathbb{S}^{n-1}}\\!\\!f(u)P_{k}^{n}(\bar{e}\cdot
u)\,du,$ (2.15)

and the Funk–Hecke Theorem implies that

$\mu\ast f\sim\sum_{k=0}^{\infty}a_{k}^{n}[f]\,\pi_{k}\mu.$ (2.16)

Hence, convolution from the right induces a multiplier transformation. It is
also easy to check from (2.14) that the convolution of zonal functions and
measures is Abelian and that for all
$\mu,\tau\in\mathcal{M}(\mathbb{S}^{n-1})$ and every $f\in
C(\mathbb{S}^{n-1},\bar{e})$,

$\int\displaylimits_{\mathbb{S}^{n-1}}\\!\\!(\mu\ast
f)(u)\,d\tau(u)=\int\displaylimits_{\mathbb{S}^{n-1}}\\!\\!(\tau\ast
f)(u)\,d\mu(u).$ (2.17)

###### Examples 2.1

  1. (a)

The cosine transform $\mathrm{C}:\mathcal{M}(\mathbb{S}^{n-1})\rightarrow
C(\mathbb{S}^{n-1})$ is defined by

$\mathrm{C}\mu(u)=\int\displaylimits_{\mathbb{S}^{n-1}}\\!\\!|u\cdot
v|\,d\mu(v)=(\mu\ast|\bar{e}\cdot\,.\,|)(u),\qquad u\in\mathbb{S}^{n-1}.$
(2.18)

Using the Formula of Rodrigues, the multipliers
$a_{k}^{n}[\mathrm{C}]:=a_{k}^{n}[|\bar{e}\cdot\,.\,|]$ of the cosine
transform can be easily computed to

$a_{k}^{n}[\mathrm{C}]=(-1)^{\frac{n-2}{2}}2\frac{1\cdot
3\cdots(k-3)}{(n+1)(n+3)\cdots(k+n-1)}$ (2.19)

for even $k$ and $a_{k}^{n}[\mathrm{C}]=0$ for $k$ odd.

  2. (b)

Generalizing (a), we define for an arbitrary body of revolution
$L\in\mathcal{K}^{n}$ the integral transform
$\mathrm{T}_{\\!L}:\mathcal{M}(\mathbb{S}^{n-1})\rightarrow
C(\mathbb{S}^{n-1})$ by

$\mathrm{T}_{L}\sigma=\sigma\ast h_{L}.$

We denote its multipliers by $a_{k}^{n}[L]:=a_{k}^{n}[h_{L}]$.

Finally, recall that the second order differential operator $\Box_{n}$,
defined by

$\Box_{n}h=h+\frac{1}{n-1}\Delta_{\mathbb{S}}h$

for $h\in C^{2}(\mathbb{S}^{n-1})$, relates the support function $h_{K}$ of a
convex body $K\in\mathcal{K}^{n}$ with its first-order area measure
$S_{1}(K,\cdot)$ by

$\Box_{n}h_{K}=S_{1}(K,\cdot),$ (2.20)

where (2.20) has to be understood in a distributional sense if $h_{K}$ is not
of class $C^{2}$. From the definition of $\Box_{n}$ and (2.7), we see that for
$h\in C(\mathbb{S}^{n-1})$ and every $k\geq 0$,

$\pi_{k}\Box_{n}h=\frac{(1-k)(k+n-1)}{n-1}\pi_{k}h.$ (2.21)

In particular, $\Box_{n}$ acts as a multiplier transformation and since such
operators clearly commute, we note for later quick reference that, by (2.16)
and (2.21), we have

$\mathrm{T}_{f}\,\Box_{n}=\Box_{n}\mathrm{T}_{f}.$ (2.22)

The $\mathcal{U}_{\alpha}$ classes. For a bounded real-valued measurable
function $f$ on $\mathbb{S}^{n-1}$ let $\|f\|_{\mathcal{U}_{\alpha}}$ denote
the smallest constant $M$ such that $\|f\|_{\infty}\leq M$ and, such that for
all positive integers $k$, there exists a polynomial $p_{k}$ of degree $k$
such that $\|f-p_{k}\|_{L^{2}}\leq Mk^{-\alpha}$. We define the class of real-
valued functions $\mathcal{U}_{\alpha}$ to be the subset of bounded functions
$f$ such that $\|f\|_{\mathcal{U}_{\alpha}}<\infty$. Let
$\Theta:[0,\infty)\to[0,1]$ be an infinitely smooth function on such that
$\Theta=1$ on $[0,1]$ and $\theta=0$ on $[2,\infty)$. Define the multiplier
operator $\mathrm{M}_{k}$ by

$\mathrm{M}_{j}f=\sum_{k=0}^{\infty}\Theta(k/j)\,\,\pi_{k}f.$ (2.23)

Note that $\mathrm{M}_{j}$ is a polynomial of degree at most $2j$ and that
$\mathrm{M}_{j}\,p=p$ for any polynomial $p$ of degree $j$. A well known fact
from approximation theory tells us that the operators $\mathrm{M}_{j}$ are
uniformly bounded in $L_{p}$, i.e. there exist a constant $C=C(\Theta)$ such
that $\|\mathrm{M}_{j}\|_{L^{p}\to L^{p}}\leq C(\Theta)$. A complete proof of
this statement can be found in Appendix A in [11].

######  Lemma 2.2 ([11])

Let $\alpha\geq 0$. If $f\in\mathcal{U}_{\alpha}$, then

$\|f-\mathrm{M}_{j}f\|_{\mathcal{U}_{\alpha}}\leq
C\|f\|_{\mathcal{U}_{\alpha}}j^{-\alpha},$

for some $C=C_{\alpha}>0$.

######  Lemma 2.3 ([11])

Let $\alpha\geq 0$. If $f,g\in\mathcal{U}_{\alpha}$, then
$fg\in\mathcal{U}_{\alpha}$ and

$\|fg\|_{\mathcal{U}_{\alpha}}\leq
C\|f\|_{\mathcal{U}_{\alpha}}\|g\|_{\mathcal{U}_{\alpha}},$

where $C=C_{\alpha}>0$.

######  Lemma 2.4 ([11])

Let $\beta>\alpha$. For every $\delta>0$ there exists a constant
$C_{\delta,\alpha,\beta}>0$ such that $\|f\|_{\mathcal{U}_{\alpha}}\leq
C_{\delta,\alpha,\beta}\|f\|_{\infty}+\delta\|f\|_{\mathcal{U}_{\beta}}$.

######  Lemma 2.5 ([11])

There exist constants $\alpha,C_{1},C_{2}>0$ with the following properties: if
$\varphi:\mathbb{S}^{n-1}\to\mathbb{R}$ satisfies
$\|\varphi\|_{2}<\varepsilon$ for some $\varepsilon\in(0,1)$, and
$\|\varphi\|_{\mathcal{U}_{\alpha}}\leq C_{1}$, then
$\|\varphi\|_{\infty}<C_{2}\varepsilon^{\frac{4}{n+3}}$.

3\. Auxiliary Results

In this section, we gather some auxiliary results that will be needed in the
proof of our main theorem. The next proposition is a generalization of
Proposition 3.2 in [13] to area measures of all degrees.

###### Proposition 3.1

Let $i\in\\{1,\dots,n-1\\}$ and $(K_{m})_{m=1}\infty$ be a sequence of convex
bodies in $\mathcal{K}^{n-1}$ with centroid at the origin. Then,
$d_{LP}(S_{i}(K_{m},\cdot),S_{i}(K,\cdot))\to 0$ as $k\to\infty$ if and only
if $d_{H}(K_{m},K)\to 0$ as $k\to\infty$.

Proof. Suppose that there exists a convex body with centroid at the origin $K$
such that $d_{LP}(S_{i}(K_{m},\cdot),S_{i}(K,\cdot))\to 0$ as $m\to\infty$.
Since convergence in the Lévy-Prokhorov metric is equivalent to weak
convergence of measures,

$V_{i}(K_{m})=\int\displaylimits_{\mathbb{S}^{n-1}}dS_{i}(K_{m},v)\to\int\displaylimits_{\mathbb{S}^{n-1}}dS_{i}(K,v)=V_{i}(K)\quad\text{as
}m\to\infty.$

Hence, there exists a constant $C>0$ such that $V_{i}(K_{m})\leq C$ for all
$m$. By the isoperimetric inequality between consecutive intrinsic volumes,
i.e. $V_{i+1}(K)^{i}V_{i}(\mathbb{B}^{n})^{i}\leq
V_{i}(K)^{i+1}V_{i+1}(\mathbb{B}^{n})^{i}$, there exists a constant
$C^{\prime}>0$ such that $V_{i+1}(K_{m})\leq C^{\prime}$ for all $m$. Again by
the weak convergence of the area measures of order $i$, for all
$u\in\mathbb{S}^{n-1}$

$h_{\Pi_{i}K_{m}}(u)=\frac{1}{2}\int\displaylimits_{\mathbb{S}^{n-1}}|u\cdot
v|dS_{i}(K_{m},v)\to\frac{1}{2}\int\displaylimits_{\mathbb{S}^{n-1}}|u\cdot
v|dS_{i}(K,v)=h_{\Pi_{i}K}(u)$ (3.1)

as $m\to\infty$. Thus, the sequence $(h_{\Pi_{i}K_{m}})_{m=1}^{\infty}$
converges uniformly to $h_{\Pi_{i}K}$ (see [42, Theorem 1.8.12]). Since
$\Pi_{i}K$ is a full dimensional convex body with centroid at the origin and
the area measures of order $i$ have centroid at the origin, there exists a
constant $C^{\prime\prime}>0$ such that

$C^{\prime\prime}\leq
h_{\Pi_{i}K_{m}}(u)=\frac{1}{2}\int\displaylimits_{\mathbb{S}^{n-1}}|u\cdot
v|dS_{i}(K_{m},v)=\int\displaylimits_{\mathbb{S}^{n-1}}\max\\{0,u\cdot
v\\}dS_{i}(K_{m},v),$ (3.2)

for all $u\in\mathbb{S}^{n-1}$ and all $m$. On the other hand, for any $s>0$
and $u\in\mathbb{S}^{n-1}$ and $m$ with $[o,su]\subseteq K_{m}$, we have

$s\max\\{0,u\cdot v\\}=h_{[o,su]}(v)\leq h_{K_{m}}(v)\quad\text{for all
}v\in\mathbb{S}^{n-1}$ (3.3)

From (3.2) and (3.3), it follows that

$\displaystyle
C^{\prime\prime}s\leq\int\displaylimits_{\mathbb{S}^{n-1}}s\max\\{0,u\cdot
v\\}dS_{i}(K_{m},v)$
$\displaystyle\leq\int\displaylimits_{\mathbb{S}^{n-1}}h(K_{m},v)dS_{i}(K_{m},v)$
$\displaystyle=nV_{i+1}(K_{m})\leq nC^{\prime}$

Therefore, $s\leq nC^{\prime}/C^{\prime\prime}$. Since $0\in K_{m}$ for all
$m$, we conclude that, for all $m$,
$K_{m}\subseteq(nC^{\prime}/C^{\prime\prime})\mathbb{B}^{n}$. By the Blaschke
selection theorem [42, Theorem 1.8.7], every subsequence of
$(K_{m})_{m=1}^{\infty}$ has a subsequence that converges in the Hausdorff
metric. The weak convergence of the area measures and
Aleksandrov–Fenchel–Jessen theorem [42, Theorem 5.2.3] imply that such
subsequences converge to $K$ with respect to the Hausdorff metric and so
$d_{H}(K_{m},K)\to 0$ as $m\to\infty$. The proof of the reverse implication is
immediate.

$\blacksquare$

The following lemma is an extension of Lemma 3 point (3) in [11] to a larger
class of multiplier operators.

###### Lemma 3.2

Let $g$ be a zonal function on $\mathbb{S}^{n-1}$ such that
$a_{k}^{n}[g]=O(k^{-\beta})$. If $f\in\mathcal{U_{\alpha}}$, then
$\mathrm{T}_{g}f\in\mathcal{U}_{\alpha+\beta}$ and there is a constant $C_{g}$
such that $\|\mathrm{T}_{g}f\|_{\mathcal{U}_{\alpha+\beta}}\leq
C_{g}\|f\|_{\mathcal{U}_{\alpha}}$.

Proof. By Parseval’s identity and (2.23), we have that

$\displaystyle\lVert\mathrm{T}_{g}f-\mathrm{T}_{g}\mathrm{M}_{j}f\rVert^{2}_{2}$
$\displaystyle=\sum_{k=j}^{\infty}a_{k}^{n}[g]^{2}(1-\Theta(k/j))^{2}\rVert\pi_{k}f\lVert^{2}_{2}$
$\displaystyle\leq
Cj^{-2\beta}\sum_{k=j}^{\infty}(1-\Theta(k/j))^{2}\rVert\pi_{k}f\lVert^{2}_{2}$
$\displaystyle\leq Cj^{-2\beta}\lVert f-\mathrm{M}_{j}f\rVert^{2}_{2}$
$\displaystyle\leq Cj^{-2(\beta+\alpha)}\|f\|_{\alpha}^{2},$

where, in the second line, we use the fact that $\Theta(k/j)\in[0,1]$ and, in
the last inequality, we use Lemma 2.2. $\blacksquare$

The next lemma was established recently in [35].

###### Lemma 3.3

Suppose that $g\in C^{2}([-1,1])$. Define $h\in
C^{2}(\mathbb{R}^{n}\setminus\\{0\\})\cap C(\mathbb{R}^{n})$ by

$h(x)=\left\\{\begin{array}[]{ll}|x|\,g\left(\displaystyle{x\cdot\bar{e}/|x|}\right)&\mbox{
for all }x\neq o,\\\ 0&\mbox{ for }x=o.\end{array}\right.$

Then

$D^{2}h(u)=\left(g(u\cdot\bar{e})-(u\cdot\bar{e})\phi^{\prime}(u\cdot\bar{e})\right)\mathrm{p}_{u^{\bot}}+g^{\prime\prime}(u\cdot\bar{e})(\mathrm{p}_{u^{\bot}}\bar{e}\otimes\mathrm{p}_{u^{\bot}}\bar{e}),$
(3.4)

where $\mathrm{p}_{u^{\bot}}=\mathrm{Id}-u\otimes u$ denotes the orthogonal
projection onto the hyperplane $u^{\bot}$.

The following theorem is one of the main results in [35]. It will be a key
ingredient in the proof of our main theorems. It provides a spectral gap for
the multipliers of convolution transforms defined in terms of symmetric convex
bodies of revolution.

###### Theorem 3.4

Suppose that $L\in\mathcal{K}^{n}$ is origin-symmetric and $\mathrm{SO}(n-1)$
invariant. Then

$|a_{k}^{n}[L]|<\frac{a_{0}^{n}[L]}{(k-1)(n+k-1)}$

for every $k>2$ and

$|a_{2}^{n}[L]|\leq\frac{a_{0}^{n}[L]}{n+1},$

where this inequality is also strict if $L$ is of class $C^{2}_{+}$.

We will also need the asymptotic behavior of the multipliers of
$\mathrm{T}_{g}$ for a smooth function $g$. This is the content of our next
lemma.

###### Lemma 3.5

Let $m\geq 2$ be an even integer and let $g$ be a zonal function on
$\mathbb{S}^{n-1}$ of class $C^{m}$. Then

$a_{k}^{n}[g]=\mathrm{O}\\!\left(k^{-m-\frac{n-2}{2}}\right)$ (3.5)

as $k\to\infty$.

Proof. In order to show (3.5), note that, by (2.15), (2.7), and the fact that
the spherical Laplacian $\Delta_{\mathbb{S}}$ is self-adjoint,

$a_{k}^{n}[g]=\frac{(-1)^{\frac{m}{2}}}{k^{\frac{m}{2}}(k+n-2)^{\frac{m}{2}}}\int\displaylimits_{\mathbb{S}^{n-1}}\\!\\!\Delta_{\mathbb{S}}^{\frac{m}{2}}g(u)P_{k}^{n}(\bar{e}\cdot
u)\,du.$

Hence, by the Cauchy–Schwarz inequality,

$|a_{k}^{n}[g]|\leq\frac{1}{k^{m}}\sqrt{\frac{\omega_{n}}{N(n,k)}}{\left\lVert\Delta^{\frac{m}{2}}_{\mathbb{S}}g\right\rVert}_{L^{2}}<\infty.$

Consequently, by (2.6), we obtain the desired asymptotic estimate

$a_{k}^{n}[g]=\mathrm{O}\left(k^{-m-\frac{n-2}{2}}\right)\quad\mbox{as
}\,\,k\to\infty.$

$\blacksquare$

The following lemma shows that the multipliers of $\mathrm{T}_{g}$ for a given
zonal function $g$ on $\mathbb{S}^{n-1}$ can be used to compute those of its
derivatives if the dimension is high enough.

###### Lemma 3.6

Let $j$ be positive integers such that $n\geq 2(j+1)$ and $g$ be a zonal
function on $\mathbb{S}^{n-1}$ of class $C^{j}$. Then,

$a_{k}^{n}\left[g^{(\,j)}\right]=(2\pi)^{j}a_{k+j}^{n-2j}[g]$ (3.6)

for all $k\geq 0$.

Proof. First, we prove the case $j=1$. Using formula (2.10) and integration by
parts yields

$\displaystyle\frac{a_{k+1}^{n-2}[g]}{\omega_{n-3}}$
$\displaystyle=\int\displaylimits_{\\!-1}^{1}g(t)P^{n-2}_{k+1}(t)(1-t^{2})^{\frac{n-5}{2}}dt$
$\displaystyle=\frac{1}{n-3}\int\displaylimits_{\\!-1}^{1}g^{\prime}(t)P_{k}^{n}(t)(1-t^{2})^{\frac{n-3}{2}}dt=\frac{1}{n-3}\frac{a_{k}^{n}[g^{\prime}]}{\omega_{n-1}}.$

Since $\omega_{n-3}=\frac{n-3}{2\pi}\,\omega_{n-1}$, we obtain,

$a_{k}^{n}[g^{\prime}]=2\pi\,a_{k+1}^{n-2}[g].$

The general case follows by a simple inductive argument.

$\blacksquare$

Now we turn to studying the behavior of integral operators when applied to
functions of the $\mathcal{U}_{\alpha}$ classes. Later, we will see that the
hypothesis of the following lemma is satisfied by a variety of generating
functions $g$. However, we will leave this as one of the final steps of the
proof of Theorem 1.

###### Lemma 3.7

Let $n\geq 3$, $\alpha\geq 0$, and $g\in C^{2}([-1,1])$. Suppose that there
exists $\beta>0$, and $C_{g,\alpha,\beta}>0$ such that

$\|\nabla^{2}_{ij}\mathrm{T}_{g}f\|_{\mathcal{U}_{\alpha+\beta}}\leq
C_{g,\alpha,\beta}\|f\|_{\mathcal{U}_{\alpha}},\quad i,j\in\\{1,\dots,n\\}.$

If $f_{1},\dots f_{n}\in\mathcal{U}_{\alpha}$, then
$\mathrm{D}(D^{2}\mathrm{T}_{g}f_{1},\dots,D^{2}\mathrm{T}_{g}f_{n-1})\in\mathcal{U}_{\alpha+\beta}$
and there exists a constant $C^{\prime}_{g,\alpha,\beta}>0$ such that

$\rVert\mathrm{D}(D^{2}\mathrm{T}_{g}f_{1},\dots,D^{2}\mathrm{T}_{g}f_{n-1})\lVert_{U^{\alpha+\beta}}\leq
C^{\prime}_{g,\alpha,\beta}\rVert f\lVert_{U^{\alpha}}\cdots\rVert
f_{n-1}\lVert_{U^{\alpha}}$ (3.7)

Proof. First we note that for all $u\in\mathbb{S}^{n-1}$,

$\mathrm{D}(D^{2}\mathrm{T}_{g}f_{1}|_{u},\dots,D^{2}\mathrm{T}_{g}f_{n-1}|_{u})=n\mathrm{D}\left(u\otimes
u,\nabla^{2}\mathrm{T}_{g}f_{1}|_{u},\cdots,\nabla^{2}\mathrm{T}_{g}f_{n-1}|_{u}\right).$
(3.8)

The latter equality follows easily by expanding the mixed discriminant on the
right hand side according to the first entry. Hence,

$\mathrm{D}(D^{2}\mathrm{T}_{g}f_{1},\dots,D^{2}\mathrm{T}_{g}f_{n-1})=\frac{1}{(n-1)!}\sum_{\delta,\sigma\in
S_{n}}\\!(-1)^{\mathrm{sgn}(\delta)+\mathrm{sgn}(\sigma)}u_{\delta_{1}}u_{\sigma_{1}}\prod_{k=1}^{n-1}\nabla_{\delta_{2}\sigma_{2}}(\mathrm{T}_{g}f_{k}).$

By assumption $\|\nabla_{ij}\mathrm{T}_{g}f\|_{\mathcal{U}_{\alpha+\beta}}\leq
C\|f\|_{\mathcal{U}_{\alpha}}$ for all $i,j\in\\{1,\dots,n-1\\}$. Thus, by the
triangle inequality and Lemma 2.3, we obtain

$\displaystyle\|\mathrm{D}(D^{2}\mathrm{T}_{g}f_{1},\dots,D^{2}\mathrm{T}_{g}f_{n-1})\|_{\mathcal{U}_{\alpha+\beta}}\\!\\!$
$\displaystyle\leq C^{\prime}\\!\\!\\!\sum_{\delta,\sigma\in
S_{n}}\left\|(-1)^{\mathrm{sgn}(\delta)+\mathrm{sgn}(\sigma)}u_{\delta_{1}}u_{\sigma_{1}}\\!\\!\prod_{k=1}^{n-1}\nabla_{\delta_{2}\sigma_{2}}(\mathrm{T}_{g}f_{k})\right\|_{\mathcal{U}_{\alpha+\beta}}$
$\displaystyle\leq
C^{\prime\prime}\max_{i,j}\prod_{k=1}^{n-1}\|\nabla_{ij}\mathrm{T}_{g}f_{k}\|_{\mathcal{U}_{\alpha+\beta}}$
$\displaystyle\leq
C^{\prime\prime\prime}\prod_{k=^{1}}^{n-1}\|f_{k}\|_{\mathcal{U}_{\alpha}}.$

where $C^{\prime},C^{\prime\prime},C^{\prime\prime\prime}>0$ are constant
depending only on $g$, $\alpha,\beta$.

$\blacksquare$

Next we prove some continuity properties of the multilinear map defined by
$(f_{1},\dots,f_{n-1})\to\mathrm{D}(D^{2}\mathrm{T}_{L}f_{1},\dots,D^{2}\mathrm{T}_{L}f_{n-1})$
for $f_{1},\dots,f_{n-1}\in L^{2}(\mathbb{S}^{n-1})$.

###### Lemma 3.8

Let $n\geq 3$, and $L$ be a convex body of class $C_{+}^{2}([-1,1])$. Then,

  * $(a)$

$\|\mathrm{D}(D^{2}\mathrm{T}_{L}f_{1},\dots,D^{2}\mathrm{T}_{L}f_{n-1})\|_{2}\leq\frac{a_{0}^{n}[L]}{n-1}\,\|f_{1}\|_{2}\prod_{k=2}^{n-1}\|f_{k}\|_{\infty}$,

  * $(b)$

$\|\mathrm{D}(D^{2}\mathrm{T}_{L}f_{1},\dots,D^{2}\mathrm{T}_{L}f_{n-1})\|_{\infty}\leq(a^{n}_{0}[L])^{n-1}\prod_{k=1}^{n-1}\|f_{k}\|_{\infty},$

for all $f_{1}\dots,f_{n-1}\in C(\mathbb{S}^{n-1})$.

Proof. We normalize $T_{L}$ such that $a^{n}_{0}[L]=1$ (note that this is
always possible since $a^{n}_{0}[L]>0$). For each $v\in\mathbb{S}^{n-1}$, let
$h_{L(v)}$ be the rotated copy of $L$ with axes of revolution
$v\in\mathbb{S}^{n-1}$. Since $h_{L}\in C^{2}(\mathbb{S}^{n-1})$, it can be
easily verified that

$D^{2}\mathrm{T}_{L}f(u)=\int\displaylimits_{\mathbb{S}^{n-1}}D^{2}h_{L(v)}(u)f(v)dv,\qquad
u\in\mathbb{S}^{n-1}.$

Hence, using the linearity and continuity of the mixed discriminant in each
coordinate, we obtain

$\mathrm{D}(D^{2}\mathrm{T}_{L}f_{1},\dots,D^{2}\mathrm{T}_{L}f_{n-1})=\\!\\!\\!\\!\\!\\!\int\displaylimits_{(\mathbb{S}^{n-1})^{n-1}}\\!\\!\\!\\!\\!\\!\mathrm{D}(D^{2}h_{L(v_{1})},\dots,D^{2}h_{L(v_{n-1})})\prod_{k=1}^{n-1}f_{k}(v_{k})dv,$
(3.9)

where $dv=dv_{1}\cdots dv_{n-1}$ is the product measure. Hence, since the
mixed discriminant $\mathrm{D}(D^{2}h_{L(v_{1})},\dots,D^{2}h_{L(v_{n-1})})$
is positive,

$\displaystyle|\mathrm{D}(D^{2}\mathrm{T}_{L}f_{1},\dots,D^{2}\mathrm{T}_{L}f_{n-1})|$
$\displaystyle\leq\int\displaylimits_{(\mathbb{S}^{n-1})^{n-1}}\mathrm{D}(D^{2}h_{L(v_{1})},\dots,D^{2}h_{L(v_{n-1})})\prod_{k=1}^{n-1}|f_{k}(v_{k})|dv$
$\displaystyle\leq\prod_{k=2}^{n-1}\|f_{k}\|_{\infty}\\!\\!\\!\\!\\!\\!\\!\\!\int\displaylimits_{(\mathbb{S}^{n-1})^{n-1}}\\!\\!\\!\\!\\!\\!\mathrm{D}(D^{2}h_{L(v_{1})},\dots,D^{2}h_{L(v_{n-1})})|f_{1}(v_{1})|dv.$

Since we normalized $\mathrm{T}_{L}$ such that
$\mathrm{T}_{L}h_{\mathbb{B}^{n}}=h_{\mathbb{B}^{n}}$ and since
$D^{2}h_{\mathbb{B}^{n}}=\mathrm{Id}$, (3.9) yields

$\int\displaylimits_{(\mathbb{S}^{n-1})^{n-1}}\mathrm{D}(D^{2}h_{L(v_{1})},\dots,D^{2}h_{L(v_{n-1})})|f_{1}(v_{1})|dv=\mathrm{D}(D^{2}\mathrm{T}_{L}|f_{1}|,\mathrm{Id}[n-2])$
(3.10)

On the other hand, (3.10) together with (2.3) yields

$\int\displaylimits_{(\mathbb{S}^{n-1})^{n-1}}\mathrm{D}(D^{2}h_{L(v_{1})},\dots,D^{2}h_{L(v_{n-1})})|f_{1}(v_{1})|dv=\frac{1}{n-1}\mathrm{tr}(D^{2}T_{L}|f_{1}|)=\Box_{n}\mathrm{T}_{L}|f_{1}|,$

since for all $f\in C^{2}(\mathbb{S}^{n-1})$ we have that
$D^{2}f=\nabla^{2}_{\mathbb{S}}f+fI$ and thus
$\frac{1}{n-1}\mathrm{tr}(D^{2}f)=\frac{1}{n-1}\Delta_{\mathbb{S}}f+f$. Hence,
we have that

$|\mathrm{D}(D^{2}T_{L}f_{1},\dots,D^{2}T_{L}f_{n-1})|\leq|\Box_{n}\mathrm{T}_{L}|f_{1}||\prod_{k=2}^{n-1}\|f_{k}\|_{\infty}$
(3.11)

Therefore, taking the $L^{2}$ norm of both sides of inequality (3.11) we
obtain

$\displaystyle\|\mathrm{D}(D^{2}\mathrm{T}_{L}f_{1},\dots,D^{2}\mathrm{T}_{L}f_{n-1})\|_{2}$
$\displaystyle\leq\|\Box_{n}\mathrm{T}_{L}|f_{1}|\|_{2}\prod_{k=2}^{n-1}\|f_{k}\|_{\infty}$
$\displaystyle\leq\sup_{k\geq
0}{a_{k}^{n}[\Box_{n}T_{L}]}\|f_{1}\|_{2}\prod_{k=2}^{n-1}\|f_{k}\|_{\infty}.$

Finally, by Theorem 3.4 we have that

$\sup_{k\geq 0}{|a_{k}^{n}[\Box_{n}T_{L}]|}=\sup_{k\geq
0}{\frac{(k-1)(k+n-1)|a_{k}^{n}[L]|}{n-1}}\leq\frac{a_{0}^{n}[L]}{n-1}.$

This concludes the proof of $(a)$. Part $(b)$ follows immediately from (3.9).

$\blacksquare$

6\. Proof of the main result

In this section we prove the main results of this article. We start by showing
the global convergence of the iterations of smooth Minkowski valuations of
degree $1$. It serves as a guideline to understand the higher degree cases.

Proof of Theorem 2. Since $\Phi_{1}$ is non-trivial, we have that
$a_{0}^{n}[g]>0$. We may normalize $\Phi_{1}$ such that $a_{0}^{n}[g]=1$. Let
$K\in\mathcal{K}^{n}$ and let $\gamma=1/\pi_{0}h_{K}$. Parseval’s identity
implies that

$\|h_{\Phi_{1}\gamma
K}-h_{\mathbb{B}^{n}}\|_{2}^{2}=\sum_{k=2}^{\infty}a_{k}^{n}[\Box_{n}g]^{2}\|\pi_{k}h(\gamma
K,\cdot)\|_{2}^{2}.$ (6.1)

On the other hand, note that by (2.20), for any convex body $L$,

$h_{\Phi_{1}L}=S_{1}(L,\cdot)\ast g=\Box_{n}h_{L}\ast g=h_{L}\ast\Box_{n}g,$
(6.2)

hence, $\Box_{n}g$ is the generating function of $\Phi_{1}$ when considered as
Minkowski endomorphism. Since $\Phi_{1}$ is assumed to be monotone we have
that $f=\Box_{n}g\geq 0$ (see Theorem 1.3 part (iii) in [24]). Moreover, since
for all $k\geq 1$ and $n\geq 3$ the Legendre polynomial $P_{k}^{n}$ satisfy
$|P_{n}^{k}(t)|<1$ for all $t\in(-1,1)$ (see Lemma 2.2 in [24]),

$|a_{k}^{n}[f]|\leq\omega_{n-1}\displaystyle{\int_{-1}^{1}f(t)|P_{k}^{n}(t)|(1-t^{2})^{\frac{n-3}{2}}dt<\omega_{n-1}\displaystyle\int_{-1}^{1}f(t)(1-t^{2})^{\frac{n-3}{2}}dt=a_{0}^{n}[f]}$
(6.3)

holds for all $k\geq 2$ since $\Phi_{1}$ is non-trivial and monotone.
Moreover, by (3.6),

$a_{k}^{n}[f]=a_{k}^{n}[\Box_{n}g]=\frac{(k-1)(k+n-1)}{n-1}a_{k}^{n}[g]=\mathrm{O}(k^{-\frac{n-2}{2}})\to
0\quad\text{as }k\to\infty.$

Thus, we obtain

$\lambda_{g}=\sup_{k\geq
2}|a_{k}^{n}[f]|<a_{0}^{n}[f]=a_{0}^{n}[\Box_{n}]a_{0}^{n}[g]=1.$

Hence, inequality (6.1) yields

$\|h_{\Phi_{1}\gamma K}-h_{\mathbb{B}^{n}}\|_{2}\leq\lambda_{g}\|h_{\gamma
K}-h_{\mathbb{B}^{n}}\|_{2}$ (6.4)

Therefore, by iterating (6.4) and taking limit as $m\to\infty$ we deduce that

$\|h(\Phi_{1}^{m}\gamma
K,\cdot)-h(\mathbb{B}^{n},\cdot)\|_{2}\leq\lambda_{g}^{m}\|h(\gamma
K,\cdot)-h(\mathbb{B}^{n},\cdot)\|_{2}\to 0\quad\text{as }m\to\infty,$ (6.5)

since $\lambda_{g}\in(0,1)$. In other words,

$d_{2}(\gamma\Phi^{m}_{1}K,\mathbb{B}^{n})\to 0\quad\text{as }m\to\infty.$

Since $d_{2}$ and $d_{H}$ generate the same topology on $\mathcal{K}^{n}$ (see
Theorem 3 in [48]), we conclude that
$d_{H}(\gamma\Phi^{m}_{1}K,\mathbb{B}^{n})\to 0$ as $m\to\infty.$
$\blacksquare$

Now we prove a general theorem from which Theorem 1 will be derived easily.
From now on, we normalize $T_{L}$ such that its first multiplier is always $1$
(we can always do this since $a_{k}^{n}[L]>0$).

###### Theorem 6.1

Let $2\leq i\leq n-1$ and $\Phi_{i}:\mathcal{K}^{n}\to\mathcal{K}^{n}$ be a
continuous translation invariant Minkowski valuation of degree $i$ which
commutes with $SO(n)$ and suppose that its generating function is the support
function of a convex body of revolution $L$ satisfying

  * $(a)$

$\mathrm{T}_{L}f\in C^{2}(\mathbb{S}^{n-1})$ whenever $f\in
C(\mathbb{S}^{n-1})$,

  * $(b)$

for all $k\geq 2$, the multipliers of $T_{L}$ satisfy
$(k-1)(k+1-2)|a_{k}^{n}[L]|\leq(n-1)i^{-1}$, and

  * $(c)$

there exists $\beta>0$ such that for all $\alpha\geq 0$ and
$f\in\mathcal{U}_{\alpha}$, there exists a constant
$C_{L}=C_{L,\beta,\alpha}>0$ such that

$\|\nabla^{2}_{ij}\mathrm{T}_{L}f\|_{\mathcal{U}_{\alpha+\beta}}\leq
C_{L}\|f\|_{\mathcal{U}_{\alpha}},\quad i,j\in\\{1,\dots,n\\}.$

Then, there exists $\varepsilon>0$ such that if $K\in\mathcal{K}^{n}$ has
continuous surface area measure of order $i$ with density $s_{i}(K,\cdot)$
such that $\|1-s_{i}(\gamma K,\cdot)\|_{\infty}<\varepsilon$ for some
$\gamma>0$, then there exists a sequence of positive numbers
$(\gamma_{m})_{m=0}^{\infty}$ such that

$d_{TV}(S_{i}(\gamma_{m}\Phi_{i}^{m}K,\cdot\,),\sigma_{n-1})\to 0\quad\text{as
}\,\,m\to\infty.$ (6.6)

Before we move to the proof of Theorem 6.1, we will need to establish the
following crucial iteration lemma.

###### Lemma 6.2

Let $\alpha,C_{1},C_{2}>0$ be the constants in Lemma 2.5 and let $L$ be a body
of revolution satisfying conditions $(a)$, $(b)$, and $(c)$. There exists
$\varepsilon_{L}>0$ and $\lambda_{L}\in(0,1)$ with the following property: for
every $\varepsilon\in(0,\varepsilon_{L})$ and every function $f$ such that
$\pi_{0}f=1$, $\lVert f-1\rVert_{L^{2}}\leq\varepsilon$ and $\lVert
f-1\rVert_{U_{\alpha}}\leq C_{1}$, there exists a positive number $\gamma$
such that

$\tilde{f}=\gamma\mathrm{D}(D^{2}\mathrm{T}_{L}f[i],\mathrm{Id}[n-1-i])$

satisfies $\pi_{0}\tilde{f}=1$,
$\lVert\tilde{f}-1\rVert_{L^{2}}\leq\lambda_{L}\,\varepsilon$ and
$\lVert\tilde{f}-1\rVert_{U_{\alpha}}\leq C_{1}$.

Proof of Lemma 6.2. Fix $\varepsilon_{L}>0$. We will determine the conditions
needed for $\varepsilon_{L}>0$ to satisfy Lemma 6.2. Let
$\varepsilon\in(0,\varepsilon_{L})$ and let $f$ be a function satisfying
$\pi_{0}f=1$, $\lVert f-1\rVert_{L^{2}}\leq\varepsilon$ and $\lVert
f-1\rVert_{U_{\alpha}}\leq C_{1}$. First note that by the multilinearity of
the mixed discriminant and (2.3), we have that

$\displaystyle\mathrm{D}(D^{2}\mathrm{T}_{L}f[i],\mathrm{Id}[n-1-i])$
$\displaystyle=\mathrm{D}((D^{2}\mathrm{T}_{L}(f-1)+\mathrm{Id})[i],\mathrm{Id}[n-1-i])$
$\displaystyle=1+i\,\,\mathrm{D}(D^{2}\mathrm{T}_{L}(f-1),\mathrm{Id}[n-1])+r$
$\displaystyle=1+i\,\,\mathrm{tr}(D^{2}\mathrm{T}_{L}(f-1))+r$
$\displaystyle=1+i\,\,\Box_{n}\mathrm{T}_{L}(f-1)+r,$

where

$r=\sum_{k=2}^{i}\frac{i!}{k!(i-k)!}\mathrm{D}(D^{2}\mathrm{T}_{L}(f-1)[k],\mathrm{Id}[n-1-k]).$

Notice that by Lemma 2.5, we have that

$\|f-1\|_{\infty}<C_{2}\varepsilon^{\frac{4}{n+3}}.$ (6.7)

Hence, by Lemma 3.8 part $(a)$ and (6.7), it follows that

$\displaystyle\begin{split}\|r-\pi_{0}r\|_{L^{2}}\leq\|r\|_{L^{2}}&\leq\sum_{k=2}^{i}\frac{i!}{k!(i-k)!}\|\mathrm{D}(D^{2}\mathrm{T}_{L}(f-1)[k],\mathrm{Id}[n-1-k])\|_{L^{2}}\\\
&\leq\sum_{k=2}^{i}\frac{i!}{k!(i-k)!}\|f-1\|^{k-1}_{\infty}\|f-1\|_{L^{2}}\\\
&\leq\sum_{k=2}^{i}\frac{i!}{k!(i-k)!}C_{2}^{k-1}\varepsilon^{\frac{4(k-1)}{n+3}+1}\\\
&\leq 2^{i}C_{2}^{i-1}\varepsilon^{\frac{n+7}{n+3}},\\\ \end{split}$ (6.8)

provided that $\varepsilon_{L}<1$. In the same fashion, by applying Lemma 3.8
part $(b)$, we obtain

$\|r-\pi_{0}r\|_{\infty}\leq 2\|r\|_{\infty}\leq
2\sum_{k=2}^{i}\frac{i!}{k!(i-k)!}\|f-1\|^{k}_{\infty}\leq
2^{i+1}C_{2}^{i}\varepsilon^{\frac{8}{n+3}}.$ (6.9)

Let $\gamma=\frac{1}{1+\pi_{0}r}$, and note that

$\displaystyle\tilde{f}=\gamma\mathrm{D}(D^{2}\mathrm{T}_{L}f[i],\mathrm{Id}[n-1-i])=1+\gamma(i\,\Box\mathrm{T}_{L}(f-1)+r-\pi_{0}r).$

By Parseval’s identity,

$\displaystyle\|\Box_{n}\mathrm{T}_{L}(f-1)\|_{2}^{2}$
$\displaystyle=\sum_{k=1}^{\infty}\frac{(k-1)^{2}(k+1-2)^{2}}{(n-1)^{2}}a_{k}^{n}[L]^{2}\|\pi_{k}f\|^{2}_{2}$
$\displaystyle\leq\Lambda_{L}^{2}\sum_{k=1}^{\infty}\|\pi_{k}f\|^{2}_{2}=\Lambda_{L}^{2}\|f-1\|_{2}^{2}.$

where

$\Lambda_{L}=\sup_{k\geq 0}\frac{(k-1)(k+1-2)}{n-1}|a_{k}^{n}[L]|.$

We obtain,

$\|\Box_{n}\mathrm{T}_{L}(f-1)\|_{2}\leq\Lambda_{L}\|f-1\|_{2}.$ (6.10)

Note that, by Lemma 3.5 and condition (b) in Theorem 6.1,
$\Lambda_{L}<\frac{1}{i}.$ If $\varepsilon_{L}>0$ is such that
$2^{i+1}C_{2}^{i}\varepsilon_{L}^{\frac{8}{n+3}}\leq\frac{1}{4}$, then

$0<\gamma=\frac{1}{1+\pi_{0}r}=1-\pi_{0}r+\frac{\pi_{0}r^{2}}{1+\pi_{0}r}\leq
1+2^{i+3}C_{2}^{i}\varepsilon^{\frac{8}{n+3}},$ (6.11)

since $|\pi_{0}r|\leq\|r\|_{\infty}\leq
2^{i+1}C_{2}^{i}\varepsilon^{\frac{8}{n+3}}\leq\frac{1}{4}$. Now fix a number
$\lambda_{L}$ such that $i\Lambda_{L}<\lambda_{L}<1$. Then, using (6.8) and
(6.10) we obtain

$\displaystyle\lVert\tilde{f}-1\rVert_{2}$
$\displaystyle\leq\gamma(i\lVert\Box_{n}\mathrm{T}_{L}(f-1)\rVert_{2}+\lVert
r-\pi_{0}r\rVert_{2})$
$\displaystyle\leq(1+2^{i+3}C_{2}^{i}\varepsilon^{\frac{8}{n+3}})(i\Lambda_{L}\varepsilon+2^{i}C_{2}^{i-1}\,\,\varepsilon^{\frac{n+7}{n+3}})$
$\displaystyle\leq(i\Lambda_{L}+2^{i+3}C_{2}^{i}\varepsilon^{\frac{4}{n+3}}+2^{i}C_{2}^{i-1}\varepsilon^{\frac{8}{n+3}}+2^{2i+3}C_{2}^{2i-1}\varepsilon^{\frac{12}{n+3}})\varepsilon$
$\displaystyle\leq(i\Lambda_{L}+2^{2i+4}C_{2}^{2i-1}\varepsilon^{\frac{4}{n+3}})\varepsilon$
$\displaystyle\leq\lambda_{L}\,\varepsilon,$

provided $\varepsilon_{L}>0$ satisfies
$2^{2i+4}C_{2}^{2i-1}\varepsilon_{L}^{\frac{4}{n+3}}\leq\lambda_{L}-i\Lambda_{L}$.
On the other hand, by (6.7) and (6.9),

$\displaystyle\begin{split}\|\tilde{f}-1\|_{\infty}&\leq\gamma\,(i\,\|\Box_{n}\mathrm{T}_{L}(f-1)\|_{\infty}+\|r-\pi_{0}r\|_{\infty})\\\
&\leq\gamma\,(i\,\|f-1\|_{\infty}+\|r-\pi_{0}r\|_{\infty})\\\
&\leq(1+2^{i+3}C_{2}^{i}\varepsilon^{\frac{8}{n+3}})\,(i\,C_{2}\varepsilon^{\frac{4}{n+3}}+2^{i}C_{2}^{i-1}\varepsilon^{\frac{n+7}{n+3}})\\\
&\leq 2^{2i+5}C_{2}^{2i-1}\varepsilon^{\frac{4}{n+3}}\end{split}$ (6.12)

Finally, note that by Lemma 3.7,

$\|\tilde{f}-1\|_{\mathcal{U}_{\alpha+\beta}}\leq\|\tilde{f}\|_{\mathcal{U}_{\alpha+\beta}}+1\leq
C_{L}\|f\|_{\mathcal{U}_{\alpha}}^{i}+1\leq
C_{L}(\|f-1\|_{\mathcal{U}_{\alpha}}+1)^{i}+1\leq 4C_{L}C_{1}^{i}$

Fix $\delta>0$ such that $\delta C_{L}C_{1}^{i}\leq\frac{1}{8}C_{1}$. Then, by
(6.12) and Lemma 2.4, we obtain

$\displaystyle\|\tilde{f}-1\|_{\mathcal{U}_{\alpha}}$ $\displaystyle\leq
C_{\delta}\|\tilde{f}-1\|_{\infty}+\delta\|\tilde{f}-1\|_{\mathcal{U}_{\alpha+\beta}}$
$\displaystyle\leq
C_{\delta}2^{2i+5}C_{2}^{2i-1}\varepsilon^{\frac{4}{n+3}}+\delta
4C_{L}C_{1}^{i}\leq C_{1},$

provided that $\varepsilon_{L}>0$ is such that
$C_{\delta}2^{2i+5}C_{2}^{2i-1}\varepsilon_{L}^{\frac{4}{n+3}}\leq\frac{1}{2}C_{1}$.

$\blacksquare$

With the iteration lemma established, we may now conclude the proof of Theorem
6.1.

Proof of Theorem 6.1. Fix $\alpha$ be such as Lemma 2.5 is satisfied. Let
$K\in\mathcal{K}^{n}$ be a convex body with absolutely continuous area measure
of order $i$ such that $\|s_{i}(K,\cdot)-1\|_{\infty}<\varepsilon$ for some
$\varepsilon$ that we will fix later. Thus,
$\|s_{i}(K,\cdot)\|_{\mathcal{U}_{0}}<\varepsilon+1$. By Lemma 3.7,

$\|s_{i}(\Phi_{i}K,\cdot\,)\|_{\mathcal{U}_{\beta}}=\|\mathrm{D}(D^{2}\mathrm{T}_{L}s_{i}(K,\cdot\,)[i],\mathrm{Id}[n-1-i])\|_{\mathcal{U}_{\beta}}\leq
C_{L}\|s_{i}(K,\cdot\,)\|_{\mathcal{U}_{0}}^{i},$

and thus, for some constant $C^{\prime}_{L}>0$,

$\|s_{i}(\Phi_{i}^{m}K,\cdot\,)\|_{\mathcal{U}_{m\beta}}\leq
C^{\prime}_{L}\|s_{i}(K,\cdot\,)\|_{\mathcal{U}_{0}}^{i^{m}}\leq
C^{\prime}_{L}(1+\varepsilon)^{i^{m}}$ (6.13)

As we already pointed out in the proof of Lemma 3.8,

$D^{2}\mathrm{T}_{L}f(u)=\int\displaylimits_{\mathbb{S}^{n-1}}D^{2}h_{L(v)}(u)f(v)dv,\qquad
u\in\mathbb{S}^{n-1}.$

Using the normalization $T_{L}h_{\mathbb{B}^{n}}=h_{\mathbb{B}^{n}}$, we
obtain

$\displaystyle s_{i}(\Phi_{i}K,\cdot)$
$\displaystyle=\mathrm{D}\left(D^{2}\mathrm{T}_{L}s_{i}(K,\cdot\,)[i],\mathrm{Id}[n-1-i]\right)$
$\displaystyle=\mathrm{D}\left(D^{2}\mathrm{T}_{L}s_{i}(K,\cdot\,)[i],D^{2}\mathrm{T}_{L}1[n-1-i]\right)$
$\displaystyle=\\!\\!\\!\int\displaylimits_{(\mathbb{S}^{n-1})^{n-1}}\mathrm{D}(D^{2}h_{L(v_{1})},\dots,D^{2}h_{L(v_{n-1})})s_{i}(K,v_{1})\cdots
s_{i}(K,v_{i})dx_{1}\\!\cdots dv_{n-1}.$

Since $1-\varepsilon<s_{i}(K,u)<1+\varepsilon$ for all $u\in\mathbb{S}^{n-1}$,
we have that

$(1-\varepsilon)^{i^{m}}<s_{i}(\Phi^{m}_{i}K,u)<(1+\varepsilon)^{i^{m}},$
(6.14)

for all $u\in\mathbb{S}^{n-1}$. Thus,
$f=s_{i}(\Phi^{m}_{i}K,\cdot\,)/\pi_{0}s_{i}(\Phi^{m}_{i}K,\cdot\,)$ is such
that $\pi_{0}f=1$ and, by (6.13) and (6.14),

$\|f-1\|_{\mathcal{U}_{m\beta}}\leq
1+\frac{\|s_{i}(\Phi^{m}_{i}K,\cdot\,)\|_{\mathcal{U}_{m\beta}}}{\pi_{0}s_{i}(\Phi^{m}_{i}K,\cdot\,)}<1+\frac{C^{\prime}_{L}(1+\varepsilon)^{i^{m}}}{(1-\varepsilon)^{i^{m}}}\leq
C^{\prime}_{L}4^{i^{m}},$ (6.15)

provided that $\varepsilon<\frac{1}{2}$. Fix $m$ large enough so that
$m\beta>\alpha>0$. Clearly from (6.14), there exists a constant $C^{\prime}>0$
such that $\|f-1\|_{\infty}<C^{\prime}\varepsilon$. Fix $\delta>0$ such that
$\delta C^{\prime}_{L}4^{i^{m}}<\frac{1}{2}C_{1}$ where $C_{1}$ is the
constant in Lemma 6.2. Hence by Lemma 2.4, we obtain

$\displaystyle\|f-1\|_{\mathcal{U}_{\alpha}}$
$\displaystyle<C_{\delta}\|f-1\|_{\infty}+\delta\|f-1\|_{\mathcal{U}_{m\beta}}$
$\displaystyle<C_{\delta}C^{\prime}\varepsilon+\delta
C^{\prime}_{L}4^{i^{m}}\leq C_{1}$

provided that $\varepsilon>0$ is such that
$C_{\delta}C^{\prime}\varepsilon\leq\frac{1}{2}C_{1}$. Therefore, without loss
of generality, we may assume that $K$ satisfies the conditions of Lemma 6.2
(if not we replace $K$ by $\Phi_{i}^{m}K$ for large enough $m$). Hence, there
exists a $\gamma_{1}$ such that the body $K_{1}=\gamma_{1}\Phi_{i}K$ satisfies
the assumptions of Lemma 6.2 with $\lambda_{L}\varepsilon$ instead of
$\varepsilon$. Applying Lemma 6.2 again with $\lambda_{L}^{2}\varepsilon$, we
obtain a positive number $\xi_{2}$ such that
$K_{2}=\xi_{2}\Phi_{i}K_{1}=\xi_{2}\gamma_{1}^{i}\Phi_{i}^{2}K=\gamma_{2}\Phi_{i}^{2}K$
satisfies the assumptions of Lemma 6.2 with $\lambda_{L}^{2}\varepsilon$,
where $\gamma_{2}=\gamma_{1}\xi_{2}$. Thus, continuing this process
inductively, we find a sequence $(\gamma_{m})_{m=1}^{\infty}$ of positive
numbers such that

$\|s_{i}(\gamma_{m}\Phi^{m}_{i}K,\cdot\,)-1\|_{L^{2}}\leq\lambda_{L}^{m}\varepsilon,$

and $\|s_{i}(\gamma_{m}\Phi^{m}_{i}K,u)\|_{\mathcal{U}_{\alpha}}\leq C_{1}$.
Hence, Lemma 2.5 yields

$d_{\mathrm{TV}}(S_{i}(\gamma_{m}\Phi^{m}_{i}K,\cdot\,),\sigma_{n-1})\leq\|s_{i}(\gamma_{m}\Phi^{m}_{i}K,\cdot\,)-1\|_{\infty}\leq
C\lambda_{L}^{m}\varepsilon\to 0\quad\text{as }\,m\to\infty,$

which is the desired result. $\blacksquare$

Note that the assumption of Theorem 1 on the support function of $K$ is
stronger than the assumption of Theorem 6.1 for the area density. The area
density of order $i$ of $K$ is given in terms of a mixed discriminate
involving the Hessian of $h_{K}$ and the identity matrix. Hence, if we assume
that $h_{K}$ is close enough to $h_{\mathbb{B}^{n}}$ in the $C^{2}$ norm then,
by continuity, $s_{i}(K,\cdot)$ will be close to the surface area measure of
the sphere $s_{i}(\mathbb{B}^{n},\cdot)=\sigma_{n-1}$.

Proof of Theorem 1. By Theorem 6.1, it remains to check that $L$ satisfies the
condition $(a)$, $(b)$ and $(c)$. Since $h_{L}\in
C^{\infty}(\mathbb{S}^{n-1})$, as in the proof of Lemma 3.8, we have that

$\nabla^{2}\mathrm{T}_{L}f(u)=\int\displaylimits_{\mathbb{S}^{n-1}}\nabla^{2}h_{L(v)}(u)f(v)dv,\qquad
u\in\mathbb{S}^{n-1}.$ (6.16)

This verifies $(a)$. Condition $(b)$ of Theorem 6.1 is the content of Theorem
3.4. Finally, to verify that $(c)$ is satisfied. Note that by Lemma 3.3,

$\nabla h_{L(u)}(u)=\left(g(u\cdot v)-(u\cdot v)g^{\prime}(u\cdot
v)\right)\mathrm{p}_{u^{\bot}}+g^{\prime\prime}(u\cdot
v)(\mathrm{p}_{u^{\bot}}v\otimes\mathrm{p}_{u^{\bot}}v),$ (6.17)

where $g\in C^{\infty}([-1,1])$ is such that $h_{L(v)}(u)=g(v\cdot u)$ for all
$u,v\in\mathbb{S}^{n-1}$.Note that $\mathrm{T}_{L}=\mathrm{T}_{g}$. Thus,
using (3.3) in (6.17) yields

$\displaystyle\nabla^{2}\mathrm{T}_{g}f(v)$
$\displaystyle=\int\displaylimits_{\mathbb{S}^{n-1}}g(u\cdot
v)\mathrm{p}_{u^{\perp}}f(u)du-\int\displaylimits_{\mathbb{S}^{n-1}}g^{\prime}(u\cdot
v)(u\cdot v)\mathrm{p}_{u^{\bot}}f(u)du$
$\displaystyle\,\,\,\,\,\,+\int\displaylimits_{\mathbb{S}^{n-1}}g^{\prime\prime}(u\cdot
v)(\mathrm{p}_{u^{\bot}}v\otimes\mathrm{p}_{u^{\bot}}v)f(u)du$
$\displaystyle=\mathrm{T}_{g}[\mathrm{p}_{u^{\perp}}f(u)](v)-\mathrm{T}_{g^{\prime}}[(v\cdot
u)\,\,\mathrm{p}_{u^{\perp}}f(u)](v)+\mathrm{T}_{g^{\prime\prime}}[(\mathrm{p}_{u^{\bot}}v\otimes\mathrm{p}_{u^{\bot}}v)f(u)].$

Therefore for all $i,j\in\\{1,\dots,n-1\\}$, we can write

$\nabla_{ij}T_{g}f(v)=\mathrm{T}_{g}[(\delta_{ij}-u_{i}u_{j})f(u)]-T_{g^{\prime}}[(v\cdot
u)(\delta_{ij}-u_{i}u_{j})f(u)]\\\
+\mathrm{T}_{g^{\prime\prime}}[(v_{i}-(v\cdot u)u_{i})(v_{j}-(v\cdot
u)u_{j})f(u)]$ (6.18)

Since, for all $i,j\in\\{1,\dots,n-1\\}$, the functions
$p_{1}(u,v)=\delta_{ij}-u_{i}u_{j}$, $p_{2}(u,v)=(v\cdot
u)(\delta_{ij}-u_{i}u_{j})$, $p_{3}(u,v)=(v_{i}-(v\cdot u)u_{i})(v_{j}-(v\cdot
u)u_{j})$ are all polynomials they belong to $\mathcal{U}_{\alpha}$ for all
$\alpha>0$. Furthermore, since $g$ is a smooth function, Lemma 3.5 implies
that its multipliers and all the multipliers of all of its derivatives decay
as fast as any polynomial. Hence, for any fixed $\beta>0$ we have that
$a_{k}^{n}[g^{(i)}]=\mathrm{O}(k^{-\beta})$ as $k\to\infty$ for $i=0,1,2$.
Thus, Lemma 3.7 together with Lemma 2.3 imply that

$\|\mathrm{T}_{g^{(i)}}[p_{k}f]\|_{\mathcal{U}_{\alpha+\beta}}\leq
C\|f\|_{\mathcal{U}_{\alpha}}$ (6.19)

for all $i\in\\{0,1,2\\}$, where $C$ is a positive constant depending on
$\|v_{i}v_{j}\|_{\mathcal{U}_{\alpha}}$,
$\|v_{i}v_{j}v_{k}\|_{\mathcal{U}_{\alpha}}$,
$\|v_{i}v_{j}v_{k}v_{l}\|_{\mathcal{U}_{\alpha}}$ for $i,j,k,l\in\\{1,\dots
n\\}$ , and $L$. Hence, by (6.19) and (6.18) we obtain

$\|\nabla_{ij}T_{g}f\|_{\mathcal{U}_{\alpha+\beta}}\leq
C^{\prime}\|f\|_{\mathcal{U}_{\alpha}}$

for some positive constant $C^{\prime}$. This verifies condition $(c)$.
Therefore, we may apply Theorem 6.1 to find an $\varepsilon>0$ such that if
$K\in\mathcal{K}^{n}$ has continuous surface area measure of order $i$ with
density $s_{i}(K,\cdot)$ such that

$\|1-s_{i}(\gamma K,\cdot)\|_{\infty}<\varepsilon,$

for some $\gamma>0$, then there exists a sequence of positive numbers
$(\gamma_{m})_{m=0}^{\infty}$ such that

$d_{TV}(S_{i}(\gamma_{m}\Phi_{i}^{m}K,\cdot\,),\sigma_{n-1})\to 0\quad\text{as
}\,\,m\to\infty.$

Since convergence with respect to the total variation metric implies
convergence with respect to the $d_{LP}$-metric, and convergence with respect
to the $d_{LP}$-metric is equivalent to convergence with respect to the
Hausdorff metric (Proposition 3.1), we have that

$d_{H}(\gamma_{m}\Phi_{i}^{m}K,\mathbb{B}^{n})\to 0\quad\text{as
}\,\,m\to\infty.$

$\blacksquare$

Let us point out that it is possible to weaken the regularity assumptions on
the support function of $L$ if one assumes that the dimension is higher. In
this case we employ Lemma 3.6 to verify condition $(c)$ in Theorem 6.1.

Proof of Theorem 1 with $L$ of class $C^{2}_{+}$ and $n\geq 6$. The proof that
conditions $(a)$, $(c)$, and $(d)$ are satisfied here is essentially the same
as the proof of Theorem 1. However, to obtain condition $(b)$, note that by
Lemma 3.5,

$a_{k}^{n}[g]=\mathrm{O}\\!\left(k^{-\frac{n+2}{2}}\right)\quad\text{as
}k\to\infty.$ (6.20)

By Lemma 3.6, $a_{k}^{n}[g^{\prime}]=2\pi\ a_{k+1}^{n-2}[g]$ and
$a_{k}^{n}[g^{\prime\prime}]=(2\pi)^{2}a_{k+2}^{n-4}[g]$ for all $k\geq 0$.
Hence, by applying Lemma 3.5 again, we obtain

$a_{k}^{n}[g^{\prime}]=\mathrm{O}\\!\left(k^{-\frac{n}{2}}\right)\text{ and
}a_{k}^{n}[g^{\prime\prime}]=\mathrm{O}\\!\left(k^{-\frac{n-2}{2}}\right)\quad\text{as
}k\to\infty.$ (6.21)

Finally, using (6.18) we obtain (in the same way as in the proof of Theorem
1),

$\|\nabla_{ij}\mathrm{T}_{g}f\|_{\mathcal{U}_{\alpha+\frac{n-2}{2}}}\leq
C^{\prime}\|f\|_{\mathcal{U}_{\alpha}},$ (6.22)

for all $i,j\in\\{1,\dots,n\\}$. The rest of the proof is now identical to the
proof of Theorem 1. $\blacksquare$

Finally, we finish this article by showing how Theorem 1 can be use to derive
a local solution to Conjecture 1 via the class reduction technique
(Proposition 1.2). First we prove Corollary 1.

Proof of Corollary 1. Suppose $K$ satisfies the hypothesis of Theorem 1. By
iterating $\Phi_{i}$, we obtain

$\Phi_{i}^{2m}K=\alpha_{m}K,$

for some sequence of real numbers $\alpha_{m}$. Therefore, we can find a
sequence $(\gamma_{m})_{m=1}^{\infty}$ by Theorem 1 such that

$\gamma_{2m}\alpha_{m}K=\gamma_{2m}\Phi_{i}^{2m}K\to\mathbb{B}^{n}\quad\text{as
}m\to\infty$

in Hausdorff metric. By continuity of the mean width we derive
$\gamma_{2m}\alpha_{m}\to 2/w(K),$ and therefore,
$\gamma_{2m}\alpha_{m}K\to\frac{2}{w(K)}K$. Hence, by uniqueness of the limit,
$K=\frac{w(K)}{2}\mathbb{B}^{n}.$ $\blacksquare$

Proof of Corollary 2. Suppose that $K\in\mathcal{K}^{n}$ satisfies the
hypothesis of Theorem 1. Let $\psi:\mathcal{K}^{n}\to\mathbb{R}$ be the
functional defined by

$\psi(K)=\frac{V_{i+1}(\Phi_{i}K)}{V_{i+1}(K)^{i}},$

for all $K\in\mathcal{K}^{n}$. Since $V_{i+1}$ is homogeneous,

$\psi(\gamma_{m}\Phi_{i}^{m}K)=\psi(\Phi_{i}^{m}K)$

for all $m$. By Proposition 1.2,

$\psi(K)\geq\psi(\Phi_{i}K)\geq\cdots\geq\psi(\Phi_{i}^{m}K)\to\psi(\mathbb{B}^{n})\quad\text{as
}m\to\infty$ (6.23)

Therefore, $\psi(K)\geq\psi(\mathbb{B}^{n})$ as required. To deal with the
equality cases locally, suppose that $\psi(K)=\psi(\mathbb{B}^{n})$. The
second part of Proposition 1 implies that $\Phi_{i}^{2}K=\alpha K$. By
Corollary 1 we obtain the desire conclusion. $\blacksquare$

Acknowledgments The author was supported by the Austrian Science Fund (FWF),
Project number: P31448-N35.

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Oscar Ortega-Moreno
Vienna University of Technology
<EMAIL_ADDRESS>