# Uniqueness of solutions to some classes of anisotropic and isotropic
curvature problems

Haizhong Li  Department of Mathematical Sciences, Tsinghua University, Beijing
100084, P.R. China<EMAIL_ADDRESS>and  Yao Wan  Department of
Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China
<EMAIL_ADDRESS>
###### Abstract.

In this paper, we apply various methods to establish the uniqueness of
solutions to some classes of anisotropic and isotropic curvature problems.
Firstly, by employing integral formulas derived by S. S. Chern [Ch59], we
obtain the uniqueness of smooth admissible solutions to a class of
Orlicz-(Christoffel)-Minkowski problems. Secondly, inspired by Simon’s
uniqueness result [Si67], we then prove that the only smooth strictly convex
solution to the following isotropic curvature problem

(0.1)
$\left(\frac{P_{k}(W)}{P_{l}(W)}\right)^{\frac{1}{k-l}}=\psi(u,r)\quad\text{on}\
\mathbb{S}^{n}$

must be an origin-centred sphere, where $W=(\nabla^{2}u+ug_{0})$,
$\partial_{1}\psi\geq 0,\partial_{2}\psi\geq 0$ and at least one of these
inequalities is strict. As an application, we establish the uniqueness of
solutions to the isotropic Gaussian-Minkowski problem. Finally, we derive the
uniqueness result for the following isotropic $L_{p}$ dual Minkowski problem

(0.2) $u^{1-p}r^{q-n-1}\det(W)=1\quad\text{on}\ \mathbb{S}^{n},$

where $-n-1<p\leq-1$ and $n+1\leq q\leq
n+\frac{1}{2}+\sqrt{\frac{1}{4}-\frac{(1+p)(n+1+p)}{n(n+2)}}$. This result
utilizes the method developed by Ivaki and Milman [IM23] and generalizes a
result due to Brendle, Choi and Daskalopoulos [BCD17].

###### Key words and phrases:

Uniqueness, $L_{p}$ dual Minkowski problem, Orlicz-Minkowski problem, Orlicz-
Christoffel-Minkowski problem, anisotropic curvature problem, isotropic
curvature problem

###### 2020 Mathematics Subject Classification:

53A07; 52A20 ; 35A02; 35K96

##  1\. Introduction

Let $M_{0}$ be a smooth closed strictly convex hypersurface in Euclidean space
$\mathbb{R}^{n+1}$ which encloses the origin, and let
$X_{0}:\mathbb{S}^{n}\rightarrow\mathbb{R}^{n+1}$ be a smooth immersion with
$X_{0}(\mathbb{S}^{n})=M_{0}$. Suppose $f$ is a positive continuous function
defined on $\mathbb{S}^{n}$, $\varphi$ is a positive continuous function
defined in $(0,\infty)$, and $k$ is an integer with $1\leq k\leq n$. We
consider the anisotropic expanding curvature flow given by a family of smooth
immersions $X:\mathbb{S}^{n}\times[0,T)\rightarrow\mathbb{R}^{n+1}$ solving
the evolution equation

(1.1) $\begin{split}\left\\{\begin{array}[]{l}\frac{\partial X}{\partial
t}(\cdot,t)=\frac{1}{f(\nu)}P_{k}(\cdot,t)\varphi(\langle X,\nu\rangle)\langle
X,\nu\rangle\nu,\\\ X(\cdot,0)=X_{0}(\cdot),\end{array}\right.\end{split}$

where $\nu$ is the unit outer normal vector of $M_{t}=X(\mathbb{S}^{n},t)$,
and $P_{k}$ is the normalized $k$-th elementary symmetric function for
principal curvature radii, i.e.

$P_{k}(\cdot,t)=\frac{1}{\binom{n}{k}}\sum\limits_{i_{1}<\ldots<i_{k}}\lambda_{i_{1}}\cdots\lambda_{i_{k}},$

$\lambda_{i}$ is the principal curvature radii of $M_{t}$. Let
$(\mathbb{S}^{n},g_{0},\nabla)$ be the unit sphere equipped with its standard
metric and Levi-Civita connection. For a smooth closed strictly convex
hypersurface $M$ with the support function $u=\langle X,\nu\rangle$, we note
that principal curvature radii $(\lambda_{1},\ldots,\lambda_{n})$ of $M$ are
the eigenvalues of the positive-definite matrix $W=(\nabla^{2}u+ug_{0})$. We
call $X:\mathbb{S}^{n}\rightarrow\mathbb{R}^{n+1}$ a homothetic self-similar
solution to the flow (1.1) if it satisfies

(1.2) $c\varphi(u)P_{k}(W)=f\quad\text{on}\ \mathbb{S}^{n},$

where $c$ is a positive constant and $u$ is the support function of
$M=X(\mathbb{S}^{n})$. In this paper, we are concerned with the uniqueness of
solutions to (1.2).

When $k=n$, (1.2) can be written as a Monge-Ampère type equation

(1.3) $c\varphi(u)\det(W)=f\quad\text{on}\ \mathbb{S}^{n},$

which corresponds to the smooth case of the Orlicz-Minkowski problem. The
Orlicz-Minkowski problem is a fundamental problem in convex geometry within
the framework of the Orlicz-Brunn-Minkowski theory. It serves as a
generalization of the classical Minkowski problem, which investigates the
necessary and sufficient conditions for a Borel measure on the unit sphere
$\mathbb{S}^{n}$ to be a multiple of the Orlicz surface area measure of a
convex body in $\mathbb{R}^{n+1}$. In [HLYZ10], Haberl, Lutwak, Yang and Zhang
studied the even case of the Orlicz-Minkowski problem. Since then, this
problem has attracted significant attention from numerous scholars, see e.g.
[BBC19, BIS21, GHWXY19, GHXY20, HH12, HLYZ16, JL19, Sa22, Sun18, SL15, XJL14].
In particular, when $\varphi(s)=s^{1-p}$, (1.3) reduces to the $L_{p}$
Minkowski problem, which has been extensively studied in the literature (see,
e.g. [And03, Bor22, BLYZ13, CW06, CY76, HLW16, HLX15, HLYZ05, JLW15, JLZ16,
KM22, LW13, Lut93, Mil21, Zhu14] and Schneider’s book [Sch14] for relevant
references). In the smooth case, the $L_{p}$ Minkowski problem is equivalent
to solving the following PDE:

(1.4) $u^{1-p}\det(W)=f\quad\text{on}\ \mathbb{S}^{n}.$

When $1\leq k\leq n-1$, (1.2) is known as the Orlicz-Christoffel-Minkowski
problem, see e.g. [JLL21]. Moreover, when $1\leq k\leq n-1$ and
$\varphi(s)=s^{1-p}$, (1.2) is referred to as the $L_{p}$ Christoffel-
Minkowski problem, which can be reduced to the following nonlinear PDE:

(1.5) $u^{1-p}P_{k}(W)=f\quad\text{on}\ \mathbb{S}^{n}.$

We refer to [Ch20, GLM18, Gu10, GX18, HMS04, Iv19, SY20] as examples for
investigating the existence, uniqueness, and convexity of solutions to (1.5).

On the other hand, let $\tilde{M}_{0}$ be a smooth closed strictly convex
hypersurface in $\mathbb{R}^{n+1}$ which encloses the origin, and let
$\tilde{X}_{0}:\mathbb{S}^{n}\rightarrow\mathbb{R}^{n+1}$ be a smooth
immersion with $\tilde{X}_{0}(\mathbb{S}^{n})=\tilde{M}_{0}$. Suppose $\Psi$
is a positive continuous function defined in $(0,\infty)\times(0,\infty)$, and
$l,k$ are two integer with $0\leq l<k\leq n$. We consider the isotropic
expanding curvature flow given by a family of smooth immersions
$\tilde{X}:\mathbb{S}^{n}\times[0,T)\rightarrow\mathbb{R}^{n+1}$ solving the
evolution equation

(1.6)
$\begin{split}\left\\{\begin{array}[]{l}\frac{\partial\tilde{X}}{\partial
t}(\cdot,t)=\left(\frac{P_{k}}{P_{l}}\right)^{\frac{1}{k-l}}(\cdot,t)\Psi(\langle\tilde{X},\tilde{\nu}\rangle,|\tilde{X}|)\langle\tilde{X},\tilde{\nu}\rangle\tilde{\nu},\\\
\tilde{X}(\cdot,0)=\tilde{X}_{0}(\cdot),\end{array}\right.\end{split}$

where $\tilde{\nu}$ is the unit outer normal vector of
$\tilde{M}_{t}=\tilde{X}(\mathbb{S}^{n},t)$. We call
$\tilde{X}:\mathbb{S}^{n}\rightarrow\mathbb{R}^{n+1}$ a homothetic self-
similar solution to the flow (1.6) if it satisfies

(1.7)
$c\Psi(u,r)\left(\frac{P_{k}}{P_{l}}\right)^{\frac{1}{k-l}}(W)=1\quad\text{on}\
\mathbb{S}^{n},$

where $c$ is a positive constant, $u$ is the support function and $r$ is the
radial function of $\tilde{M}=\tilde{X}(\mathbb{S}^{n})$. When $l=0$ and
$\partial_{2}\psi=0$, then (1.7) reduces to the isotropic case of (1.2).
Moreover, there are two important cases for (1.7): First, when $l=0,\ k=n$ and
$\Psi(s,t)=s^{1-p}t^{q-n-1}$, (1.7) becomes the isotropic $L_{p}$ dual
Minkowski problem, corresponding to the following equation

(1.8) $u^{1-p}r^{q-n-1}\det(W)=1\quad\text{on}\ \mathbb{S}^{n}.$

Second, when $l=0,\ k=n$ and $\Psi(s,t)=s^{1-p}e^{-\frac{t^{2}}{2}}$, (1.7)
becomes the isotropic $L_{p}$ Gaussian-Minkowski problem, corresponding to the
following equation

(1.9) $u^{1-p}e^{-\frac{r^{2}}{2}}\det(W)=1\quad\text{on}\ \mathbb{S}^{n}.$

We are also concerned with the uniqueness of solutions to isotropic curvature
problems. When $\Psi(u,r)=u^{\alpha}$, we refer to [CG21, GLW22, GLM18]
regarding the uniqueness of solutions to (1.7).

###  1.1. Uniqueness of solutions to Orlicz-(Christoffel)-Minkowski problems

In this paper, we always assume that all hypersurfaces $M^{n}$ are smooth
closed hypersurfaces in $\mathbb{R}^{n+1}$ that contain the origin in its
interior and bound a convex body $K$ with strictly positive Gauss curvature.
Denote by $u$ the support function of $K$, and by $r$ the radial function of
$K$. We write the Hessian matrix of $u$ with respect to
$(\mathbb{S}^{n},g_{0},\nabla)$ as $W=(\nabla^{2}u+ug_{0})$.

Let $\varphi:(0,\infty)\to(0,\infty)$ be a continuous function. We consider
the following Orlicz-Christoffel-Minkowski problem:

(1.10) $\varphi(u)P_{k}(W)=f\quad\text{on}\ \mathbb{S}^{n}.$

A solution $u$ to (1.10) is called admissible if
$W=(\nabla^{2}u+ug_{0})\in\Gamma_{k}$, where $\text{Sym}(n)$ is the space of
$n\times n$ symmetric matrices and $\Gamma_{k}=\\{A\in\text{Sym}(n):\
P_{i}(A)>0,\ i=1,\ldots,k\\}$.

Using integral formulas, S. S. Chern [Ch59] proved the classical Alexandrov-
Fenchel-Jessen theorem, which establishes the uniqueness of solutions to
(1.10) with $\varphi\equiv 1$. Furthermore, we obtain the following uniqueness
result by utilizing integral formulas.

######  Theorem 1.1.

Let $2\leq k\leq n$ be an integer. Let $\varphi:(0,\infty)\to(0,\infty)$ be a
continuous function that satisfies

(1.11)
$\begin{split}&(s\varphi^{\frac{1}{k}}(s)-t\varphi^{\frac{1}{k}}(t))(\varphi^{\frac{k-1}{k}}(s)-\varphi^{\frac{k-1}{k}}(t))<0,\end{split}$

for any $t>s>0$. For any positive function $f\in C(\mathbb{S}^{n})$, then the
smooth admissible solution to (1.10) is unique.

In particular, when considering the $L_{p}$ Christoffel-Minkowski problem
(1.5), the uniqueness of admissible solutions was established by Guan-Ma-
Trudinger-Zhu [Gu10] for $p=1$, by Hu-Ma-Shen [HMS04] for $p\geq k+1$, and
Guan-Xia [GX18] for $1<p<k+1$. In the isotropic case, Chen [Ch20] proved the
uniqueness of strictly convex solutions to (1.5) with $f\equiv 1$ for
$1>p>1-k$, while McCoy [Mc11] established the uniqueness for $p=1-k$. As an
application of Theorem 1.1, by choosing $\varphi(s)=s^{1-p}$, we provide a new
proof for the following uniqueness result.

######  Corollary 1.1 ([GX18]).

Let $2\leq k\leq n$ be an integer. Suppose $1<p<k+1$ is a real number. For any
positive function $f\in C(\mathbb{S}^{n})$, then the smooth admissible
solution to (1.5) is unique.

Here are some more examples of functions that satisfy the condition (1.11):

  1. (1)

$\varphi(s)=\mu s^{\alpha}+s^{\beta}$, where $0\geq\alpha>\beta>-k$ and
$\mu\geq 0$.

  2. (2)

$\varphi(s)=(\mu s^{\alpha}+s^{\beta})^{-1}$, where $0\leq\alpha<\beta<k$ and
$\mu\geq 0$.

  3. (3)

$\varphi(s)=e^{-s}$ with the assumption $0<u<k$.

  4. (4)

$\varphi(s)=s^{1-p_{1}}$, if $0<s\leq 1$; $\varphi(s)=s^{1-p_{2}}$, if $s>1$,
where $1<p_{1},p_{2}<k+1$.

###  1.2. Uniqueness of solutions to a class of isotropic curvature problems

Let $\psi:(0,\infty)\times(0,\infty)\to(0,\infty)$ be a continuous function,
and $l,k$ be two integers with $0\leq l<k\leq n$. We consider the following
isotropic curvature problem:

(1.12)
$\left(\frac{P_{k}(W)}{P_{l}(W)}\right)^{\frac{1}{k-l}}=\psi(u,r)\quad\text{on}\
\mathbb{S}^{n}.$

After giving new integral formulas involving the radial function, we obtain
the following theorem:

######  Theorem 1.2.

Let $n\geq 2$ and $0\leq l<k\leq n$. Suppose
$\psi:(0,\infty)\times(0,\infty)\to(0,\infty)$ is a continuous function, and
there exists a $C^{1}$ decreasing function $\eta:(0,\infty)\to(0,\infty)$
satisfying

(1.13)
$\displaystyle\left(s\psi(1,1)-\psi(s,t)\right)\left(\eta(1)\psi^{k-l-1}(1,1)-\eta(t)\psi^{k-l-1}(s,t)\right)\leq
0,$

for any $t>s>0$. Then the smooth strictly convex solution $M$ to (1.12) must
be a sphere. Moreover, if in addition $\eta^{\prime}(t)<0$ for any $t>0$, then
$M$ is an origin-centred sphere.

Let $\psi(u,r)=u^{\alpha}\phi(r)$, we consider the following equation

(1.14)
$\left(\frac{P_{k}(W)}{P_{l}(W)}\right)^{\frac{1}{k-l}}=u^{\alpha}\phi(r)\quad\text{on}\
\mathbb{S}^{n}.$

Note that if $0\leq\alpha<1$ and $\phi$ is increasing, then $\psi(u,r)$
satisfies the condition (1.13) with
$\eta(t)=\phi^{-\frac{k-l-1}{1-\alpha}}(t)$. Therefore, we obtain

######  Corollary 1.2.

Let $n\geq 2$ and $0\leq l<k\leq n$. Suppose $0\leq\alpha<1$ is a real number,
and $\phi:(0,\infty)\to(0,\infty)$ is a $C^{1}$ increasing function. Then the
smooth strictly convex solution $M$ to (1.14) must be a sphere. Moreover, if
in addition $\phi^{\prime}(t)>0$ for any $t>0$, then $M$ is an origin-centred
sphere.

On the other hand, Simon [Si67] studied (1.12) when $\partial_{2}\psi=0$ and
proved some uniqueness theorems, including the classical Alexandrov-Fenchel-
Jessen theorem. Inspired by Simon’s result, we obtain the following uniqueness
theorem for (1.12):

######  Theorem 1.3.

Let $n\geq 2$ and $0\leq l<k\leq n$. Suppose
$\psi:(0,\infty)\times(0,\infty)\to(0,\infty)$ is a $C^{1}$ function with
$\partial_{1}\psi\geq 0,\ \partial_{2}\psi\geq 0$ and at least one of these
inequalities is strict. Then the smooth strictly convex solution $M$ to (1.12)
must be an origin-centred sphere.

When $l=0$, we consider the following isotropic curvature problem:

(1.15) $P_{k}(W)=\psi(u,r)\quad\text{on}\ \mathbb{S}^{n}.$

Then the following corollary is a direct consequence of Theorem 1.3.

######  Corollary 1.3.

Let $n\geq 2$ and $1\leq k\leq n$. Suppose
$\psi:(0,\infty)\times(0,\infty)\to(0,\infty)$ is a $C^{1}$ function with
$\partial_{1}\psi\geq 0,\ \partial_{2}\psi\geq 0$ and at least one of these
inequalities is strict. Then the smooth strictly convex solution $M$ to (1.15)
must be an origin-centred sphere.

In particular, when $l=0$ and $k=n$, we provide a new proof of the uniqueness
result obtained by Ivaki [Iv23] for a large class of Minkowski type problems
in the case $n\geq 2$.

######  Corollary 1.4 ([Iv23]).

Let $n\geq 2$. Suppose $\psi:(0,\infty)\times(0,\infty)\to(0,\infty)$ is a
$C^{1}$ function with $\partial_{1}\psi\geq 0,\ \partial_{2}\psi\geq 0$ and at
least one of these inequalities is strict. Then the smooth strictly convex
solution $M$ to the following isotropic Minkowski type problem

(1.16) $\det(W)=\psi(u,r)\quad\text{on}\ \mathbb{S}^{n}$

must be an origin-centred sphere.

As applications, we can immediately deduce the following corollaries:

######  Corollary 1.5.

Let $n\geq 2$ and $1\leq k\leq n$. Suppose $p\geq 1,\ q\leq k+1$ and at least
one of these inequalities is strict. Then the smooth strictly convex solution
$M$ to the following isotropic $L_{p}$ dual Christoffel-Minkowski problem

(1.17) $u^{1-p}r^{q-k-1}P_{k}(W)=1\quad\text{on}\ \mathbb{S}^{n}$

must be an origin-centred sphere.

######  Remark 1.1.

When $k=n$, Corollary 1.5 reduces to the uniqueness theorem of the isotropic
$L_{p}$ dual Minkowski problem (1.8) for $p>1$ and $q\leq n+1$, as shown by
Chen-Li [CL21].

######  Corollary 1.6.

Let $n\geq 2$ and $1\leq k\leq n$. Suppose $p\geq 1$ is a real number. Then
the smooth strictly convex solution $M$ to the following isotropic $L_{p}$
Gaussian Christoffel-Minkowski problem

(1.18) $u^{1-p}e^{-\frac{r^{2}}{2}}P_{k}(W)=1\quad\text{on}\ \mathbb{S}^{n}$

must be an origin-centred sphere.

######  Remark 1.2.

Corollary 1.6 provides an affirmative answer to the conjecture proposed by
Chen-Hu-Liu-Zhao [CHLZ23], and another proof can be found in [Iv23].

###  1.3. Uniqueness of solutions to $L_{p}$ dual Minkowski problems

The $L_{p}$ dual Minkowski problem was posed by Lutwak, Yang and Zhang in
[LYZ18], and unifies the $L_{p}$ Minkowski problem and the dual Minkowski
problem. The $L_{p}$ dual Minkowski problem has been intensively studied by
many authors, see e.g. [BF19, CCL21, CHZ19, CL21, HZ18, LW22, LLL22, SX21]. In
the smooth case, given a positive continuous function $f$ on $\mathbb{S}^{n}$,
the $L_{p}$ dual Minkowski problem becomes the following Monge-Ampère type
equation:

(1.19) $u^{1-p}r^{q-n-1}\det(W)=f\quad\text{on}\ \mathbb{S}^{n}.$

For a class of convex bodies that are close to the ball, by applying integral
formulas involving the radial function, we obtain the uniqueness of solutions
to $L_{p}$ dual Minkowski problems for $p=2$ and $q\leq n+1$.

######  Theorem 1.4.

Let $n\geq 2$. Suppose $p=2$ and $q\leq n+1$. For any positive function $f\in
C(\mathbb{S}^{n})$, then the smooth strictly convex solution $M$ to (1.19)
satisfying $\frac{r}{u}\leq\sqrt{2}$ is unique, except when $p=q$, in which
case it is unique up to scaling.

######  Remark 1.3.

If a convex body $K$ satisfies $RB^{n+1}\subseteq K\subseteq\sqrt{2}RB^{n+1}$
for some $R>0$, where $B^{n+1}$ denotes the Euclidean unit ball in
$\mathbb{R}^{n+1}$, then its boundary $M=\partial K$ satisfies
$\frac{r}{u}\leq\sqrt{2}$. Consequently, the uniqueness of the $L_{2}$ dual
Minkowski problem with $q\leq n+1$ in $\mathbb{R}^{n+1}$ is guaranteed for
these convex bodies.

When $f\equiv 1$ and $q=n+1$, (1.19) reduces to the isotropic $L_{p}$
Minkowski problem:

(1.20) $u^{1-p}\det(W)=1\quad\text{on}\ \mathbb{S}^{n}$

It has been established, due to the works of Lutwak [Lut93], Andrews [And99],
Andrews-Guan-Ni [AGN16], Choi-Daskalopoulos [CD16], as well as Brendle-Choi-
Daskalopoulos [BCD17], that for $p>-n-1$, the only solutions $M$ to (1.20) are
origin-centered spheres, while for $p=-n-1$, the solutions $M$ are origin-
centered ellipsoids. Moreover, Saroglou [Sa22] obtained uniqueness results for
a class of isotropic Orlicz-Minkowski problems, including (1.20) for $p>-n-1$.
By using a local version of the Brunn-Minkowski inequality, Ivaki and Milman
[IM23] provided a new proof of the uniqueness of solutions to (1.20) for
$-n-1\leq p\leq-1$ and $p=0$.

As for the uniqueness of solutions to the isotropic $L_{p}$ dual Minkowski
problem (1.8), it is known that the solution is unique when $q<p$ [HZ18] and
is unique up to scaling when $p=q$ [CL21], which can be obtained by the strong
maximum principle. When $1<p<q\leq n+1$ or $-n-1\leq p<q<-1$, Chen-Li [CL21]
showed that the solution must be an origin-centred sphere. In the planar case
$n=1$, we [LW22] provided a complete classification of solutions to (1.8).
When $-n-1\leq p<q\leq\min\\{n+1,n+1+p\\}$, Chen-Huang-Zhao [CHZ19] proved
that the origin-symmetric solution must be an origin-centred sphere.
Furthermore, Ivaki and Milman [IM23] proved the uniqueness of origin-symmetric
solutions to (1.8) for $p\geq-n-1$ and $q\leq n+1$ with at least one of these
being strict.

In this paper, by selecting test functions in a local version of the Brunn-
Minkowski inequality, we obtain some new uniqueness results for the isotropic
$L_{p}$ dual Minkowski problem.

######  Theorem 1.5.

Let $n\geq 1$. Suppose either

  1. (i)

$p\geq-n-1,\ q\leq n-1+2\sqrt{1+\frac{n+1+p}{n+2}}$ and at least one of these
is strict, or

  2. (ii)

$q\leq n+1,\ p\geq-n+1-2\sqrt{1+\frac{n+1-q}{n+2}}$ and at least one of these
is strict.

Then the smooth strictly convex origin-symmetric solution $M$ to (1.8) must be
an origin-centred sphere.

######  Remark 1.4.

In fact, the uniqueness of origin-symmetric solutions to the isotropic $L_{p}$
Christoffel-Minkowski problem (1.17) was also investigated in [IM23]. By
modifying the test functions in [IM23, Lemma 3.1] for $k<n$, we obtain some
results similar to Theorem 1.5, see Appendix A.

When $p\geq-n-1$ and $q\leq n+1$, Theorem 1.5 reduces to the following result
of Ivaki and Milman.

######  Corollary 1.7 ([IM23]).

Let $n\geq 1$. Suppose $p\geq-n-1$, $q\leq n+1$ and at least one of these
inequalities is strict. Then the smooth strictly convex origin-symmetric
solution $M$ to the isotropic $L_{p}$ dual Minkowski problem (1.8) must be an
origin-centred sphere.

In the planar case $n=1$, combining Theorem 1.5 and [LW22, Theorem 1.2]
immediately yields the following corollary.

######  Corollary 1.8.

Let $n=1$. If $q>2$, $p+3<q\leq 2\sqrt{\frac{5+p}{3}}$; or $p<-2$,
$-2\sqrt{\frac{5-q}{3}}\leq p<q-3$, then the embedded solution to the planar
isotropic $L_{p}$ dual Minkowski problem is unique.

As for the case without the origin-symmetric assumption, we have the following
result.

######  Theorem 1.6.

Let $n\geq 1$. Suppose either

  1. (i)

$-n-1<p\leq-1$, $n+1\leq q\leq
n+\frac{1}{2}+\sqrt{\frac{1}{4}-\frac{(1+p)(n+1+p)}{n(n+2)}}$, or

  2. (ii)

$1\leq q<n+1$,
$-n-\frac{1}{2}-\sqrt{\frac{1}{4}-\frac{(1-q)(n+1-q)}{n(n+2)}}\leq p\leq-n-1$.

Then the smooth strictly convex solution $M$ to (1.8) must be an origin-
centred sphere.

When $q=n+1$, Theorem 1.6 reduces to the following uniqueness result for the
isotropic $L_{p}$ Minkowski problem (1.20) within the range $-n-1<p\leq-1$,
originally established by Brendle, Choi and Daskalopoulos [BCD17].

######  Corollary 1.9 ([BCD17, IM23, Sa22]).

Let $n\geq 1$. Suppose $-n-1<p\leq-1$. Then the smooth strictly convex
solution $M$ to the isotropic $L_{p}$ Minkowski problem (1.20) must be an
origin-centred sphere.

The paper is organized as follows. In Section 2, we collect some basic
concepts and techniques. In Section 3, we prove Theorem 1.1, Theorem 1.2 and
Theorem 1.4 by use of integral formulas. In Section 4, inspired by Simon’s
result [Si67], we give the proof of Theorem 1.3. In Section 5, using the
method developed by Ivaki and Milman [IM23], we prove Theorem 1.5 and Theorem
1.6. In Appendix A, we provide a uniqueness result of origin-symmetric
solutions to the isotropic $L_{p}$ dual Christoffel-Minkowski problem.

######  Acknowledgments.

The work was supported by NSFC Grant No. 11831005.

##  2\. Preliminaries

###  2.1. Inequalities of elementary symmetric functions

We first review some properties of elementary symmetric functions, see e.g.
[CNS85, Ni00, Ga59].

######  Definition 2.1.

For an integer $k=1,\ldots,n$ and a point
$\lambda=(\lambda_{1},\ldots,\lambda_{n})\in\mathbb{R}^{n}$, the normalized
$k$-th elementary symmetric function $P_{k}(\lambda)$ is defined by

$\displaystyle
P_{k}(\lambda)=\binom{n}{k}^{-1}\sigma_{k}(\lambda)=\binom{n}{k}^{-1}\sum_{1\leq
i_{1}<\cdots<i_{k}\leq n}\lambda_{i_{1}}\cdots\lambda_{i_{k}}.$

It is convenient to set $P_{0}(\lambda)=1$ and $P_{k}(\lambda)=0$ for $k>n$.

The definition can be extended to symmetric matrices. Let
$A\in\mathrm{Sym}(n)$ be an $n\times n$ symmetric matrix and
$\lambda=\lambda(A)$ be the eigenvalues of $A$. Set
$P_{k}(A)=P_{k}(\lambda(A))$. We have

$\displaystyle P_{k}(A)=$
$\displaystyle\frac{(n-k)!}{n!}\delta_{i_{1},\ldots,i_{k}}^{j_{1},\ldots,j_{k}}A_{i_{1}j_{1}}\cdots
A_{i_{k}j_{k}},\quad k=1,\ldots,n,$

where
$\delta_{i_{1},\ldots,i_{k}}^{j_{1},\ldots,j_{k}}=\det(\delta_{i_{s}}^{j_{t}})_{k\times
k}$ denotes the generalized Kronecker delta.

######  Lemma 2.2.

If $\lambda\in\Gamma_{k}=\\{\lambda\in\mathbb{R}^{n}:\ P_{i}(\lambda)>0,\
i=1,\ldots,k\\}$, we have the following Newton-MacLaurin inequality

(2.1) $\displaystyle P_{k}(\lambda)P_{l-1}(\lambda)\leq
P_{l}(\lambda)P_{k-1}(\lambda),\quad 1\leq l<k.$

Equality holds if and only if $\lambda_{1}=\ldots=\lambda_{n}$.

Furthermore, we state the following properties which can be found in [Ni00,
Theorem 2.1].

######  Lemma 2.3 ([Ni00]).

Suppose that $\alpha,\beta\in(0,1)$ and $k,l\in\mathbb{N}$ are numbers such
that

$\displaystyle\alpha+\beta=1\quad\text{and}\quad
k\alpha+l\beta\in\\{0,\ldots,n\\}.$

If $\lambda\in\Gamma_{k}\cap\Gamma_{l}$, we have

(2.2) $\displaystyle P_{k\alpha+l\beta}(\lambda)\geq
P_{k}^{\alpha}(\lambda)P_{l}^{\beta}(\lambda).$

Equality holds if and only if $\lambda_{1}=\ldots=\lambda_{n}$.

######  Definition 2.4.

For an integer $k=1,\ldots,n$ and $k$ points
$\lambda^{(1)},\ldots,\lambda^{(k)}\in\mathbb{R}^{n}$, the normalized $k$-th
mixed elementary symmetric function
$P_{k}(\lambda^{(1)},\ldots,\lambda^{(k)})$ is defined by

$\displaystyle
P_{k}(\lambda^{(1)},\ldots,\lambda^{(k)})=\binom{n}{k}^{-1}\sigma_{k}(\lambda^{(1)},\ldots,\lambda^{(k)})=\binom{n}{k}^{-1}\sum_{1\leq
i_{1}<\cdots<i_{k}\leq n}\lambda_{i_{1}}^{(1)}\cdots\lambda_{i_{k}}^{(k)}.$

The definition can also be extended to symmetric matrices. Let
$A^{1},\ldots,A^{k}\in\mathrm{Sym}(n)$ be $n\times n$ symmetric matrices and
$\lambda^{(i)}=\lambda(A^{i})$ be the eigenvalues of $A^{i}$ for
$i=1,\ldots,k$. Set
$P_{k}(A^{1},\ldots,A^{k})=P_{k}(\lambda^{(1)},\ldots,\lambda^{(k)})$. We have

$\displaystyle P_{k}(A^{1},\ldots,A^{k})$
$\displaystyle=\frac{(n-k)!}{n!}\delta_{i_{1},\ldots,i_{k}}^{j_{1},\ldots,j_{k}}A_{i_{1}j_{1}}^{1}\cdots
A_{i_{k}j_{k}}^{k}.$

It is obvious that

$\displaystyle
P_{k}(\lambda,\ldots,\lambda)=P_{k}(\lambda)\quad\text{and}\quad
P_{n}(\lambda^{(1)},\ldots,\lambda^{(k)},I,\ldots,I)=P_{k}(\lambda^{(1)},\ldots,\lambda^{(k)}),$

equivalently,

$\displaystyle P_{k}(A,\ldots,A)=P_{k}(A)\quad\text{and}\quad
P_{n}(A^{1},\ldots,A^{k},Id,\ldots,Id)=P_{k}(A^{1},\ldots,A^{k}),$

where $I=(1,\ldots,1)\in\mathbb{R}^{n}$, and $Id$ is the $n\times n$ identity
matrix.

Since $P_{k}$ is hyperbolic and complete, we have the following lemma, see
[Ga59, Theorem 5].

######  Lemma 2.5 ([Ga59]).

For any $A^{i}\in\Gamma_{k},\ i=1,\ldots,k$, we have

(2.3) $\displaystyle P_{k}(A^{1},\ldots,A^{k})\geq
P_{k}(A^{1})^{\frac{1}{k}}\cdots P_{k}(A^{k})^{\frac{1}{k}}.$

Equality holds if and only if these $k$ matrices are pairwise proportional.

In particular, define
$P_{k,l}(\lambda,\bar{\lambda})=P_{k+l}(\underbrace{\lambda,\ldots,\lambda}_{k},\underbrace{\bar{\lambda},\ldots,\bar{\lambda}}_{l})$.
Then we have

######  Lemma 2.6.

For any $\lambda,\bar{\lambda}\in\Gamma_{k+l}$,

(2.4) $\displaystyle P_{k,l}(\lambda,\bar{\lambda})\geq
P_{k+l}(\lambda)^{\frac{k}{k+l}}P_{k+l}(\bar{\lambda})^{\frac{l}{k+l}},$

Equality holds if and only if $\lambda$ and $\bar{\lambda}$ are proportional.

###  2.2. $L_{k}$ operator

Let $x:M^{n}\to\mathbb{R}^{n+1}$ be a hypersurface, and choose a unit normal
vector $e_{n+1}$ such that $\\{e_{1},\ldots,e_{n},e_{n+1}\\}$ is a local
orthonormal basis. Let $(h_{ij})$ be the second fundamental form and
$\sigma_{k}(h_{ij})$ be the $k$-th mean curvature of $M$. Denote by
$u=-\langle x,e_{n+1}\rangle$ the support function and by
$P_{k}(h_{ij})=\binom{n}{k}^{-1}\sigma_{k}(h_{ij})$ the normalized $k$-th mean
curvature of $M$.

######  Definition 2.7.

For a hypersurface $x:M^{n}\to\mathbb{R}^{n+1}$, the $k$-th Newton tensor is
defined by

$\displaystyle
T_{ij}^{k}=\sigma_{k}(h_{ij})\delta_{ij}-\sigma_{k-1}(h_{ij})h_{ij}+\ldots+(-1)^{k}\sum\limits_{i_{1},\ldots,i_{k}}h_{ii_{1}}h_{i_{1}i_{2}}\cdots
h_{i_{k}j},\quad k=0,\ldots,n-1.$

We have a second-order differential operator $L_{k}$ defined by

$\displaystyle L_{k}f=\sum_{i,j}T_{ij}^{k}f_{ij},\quad\forall\ f\in
C^{\infty}(M),\ 0\leq k\leq n-1.$

$L_{k}$ operator satisfies the following properties, see [Re73, HL08]:

######  Proposition 2.8.

  1. (i)

$T_{ij}^{k}=T_{ji}^{k},\ \sum_{j}T_{ij,j}^{k}=0$ and
$T_{ij}^{k}=\frac{1}{k!}\delta_{i_{1},\ldots,i_{k},i}^{j_{1},\ldots,j_{k},j}h_{i_{1}j_{1}}\cdots
h_{i_{k}j_{k}}$.

  2. (ii)

$L_{k}$ operator is elliptic and self-adjoint for a strictly convex
hypersurface $M$.

  3. (iii)

We have

(2.5) $\displaystyle\frac{1}{2}L_{k}(|x|^{2})=\sum T_{ij}^{k}(\langle
x,e_{i}\rangle)_{j}=(n-k)\binom{n}{k}(P_{k}(h_{ij})-uP_{k+1}(h_{ij})).$

###  2.3. Integral formulas for hypersurfaces

In this subsection, we review some integral formulas derived by S. S. Chern
[Ch59] and provide some new integral formulas that contain radial functions.

Let $x:M^{n}\to\mathbb{R}^{n+1}$ and $\bar{x}:\bar{M}^{n}\to\mathbb{R}^{n+1}$
be two closed strictly convex smooth hypersurfaces such that the normal
vectors of $M$ at $x$ and of $\bar{M}$ at $\bar{x}$ are the same. Let
$\\{e_{1},\ldots,e_{n},e_{n+1}\\}$ be an orthonormal frame in
$\mathbb{R}^{n+1}$ such that $e_{n+1}$ is the unit inner normal vector of $M$
at $x$, and also the unit inner normal vector of $\bar{M}$ at $\bar{x}$. We
will denote the following ranges of indices:

$1\leq A,B,\ldots\leq n+1;\quad 1\leq i,j,\ldots\leq n.$

Let $\\{\theta_{A}\\}$ be the dual frame field of $\\{e_{A}\\}$, and denote
$\omega_{A}=\left.\theta_{A}\right|_{M}$ and
$\bar{\omega}_{A}=\left.\theta_{A}\right|_{\bar{M}}$. Restricting $\theta_{A}$
to the hypersurface $M$, we have $\theta_{i}=\omega_{i}$ and $\theta_{n+1}=0$.
We have

$dx=\sum\limits_{i}\omega_{i}e_{i},\quad
de_{n+1}=\sum\limits_{i}\omega_{n+1,i}e_{i},$

where $\omega_{i,n+1}=\sum\limits_{j}h_{ij}\omega_{j}$. Let
$x=\sum\limits_{A}\tau_{A}e_{A}$, $(h^{ik})=(h_{ik})^{-1}$, and denote by
$u=-\langle x,e_{n+1}\rangle$ the support function and by $r=\langle
x,x\rangle^{\frac{1}{2}}$ the radial function of $M$, then we obtain

(2.6) $\displaystyle\tau_{n+1}=-u,\quad
r^{2}=\sum\limits_{i}\tau_{i}^{2}+u^{2},\quad
dr=\frac{1}{r}\sum\limits_{i}\tau_{i}\omega_{i}.$

We assign notations $\bar{\omega}_{n+1,i},\ \bar{h}_{ij},\ \bar{h}^{ij},\
\bar{\tau}_{A},\ \bar{u}$ and $\bar{r}$ for hypersurface $\bar{M}$ as well.

Notice that $\omega_{n+1,i}=\langle
de_{n+1},e_{i}\rangle=\bar{\omega}_{n+1,i}$. Let
$a_{ij}=\sum\limits_{k}h^{ik}\bar{h}_{kj}$, we have

(2.7) $\displaystyle\omega_{i}=\sum\limits_{j}a_{ij}\bar{\omega}_{j}.$

Consider the differential form of degree $n-1$:

(2.8)
$\displaystyle\Phi_{0}=(x,\bar{x},\underbrace{d\bar{x},\ldots,d\bar{x}}_{n-1}),$

where $(\cdot,\ldots,\cdot)$ is a determinant of order $n+1$, whose columns
are the components of the respective vectors or vector-valued differential
forms, with the convention that in the expansion of the determinant the
multiplication of differential forms is in the sense of exterior
multiplication. Then applying exterior differentiation gives

(2.9)
$\begin{split}d\Phi_{0}&=(x,d\bar{x},\underbrace{d\bar{x},\ldots,d\bar{x}}_{n-1})+(dx,\bar{x},\underbrace{d\bar{x},\ldots,d\bar{x}}_{n-1})\\\
&=(\sum_{i}\tau_{i}e_{i}-ue_{n+1},\sum_{i}\bar{\omega}_{i}e_{i},\underbrace{\sum_{i}\bar{\omega}_{i}e_{i},\ldots,\sum_{i}\bar{\omega}_{i}e_{i}}_{n-1})\\\
&\quad+(\sum_{i}\omega_{i}e_{i},\sum_{i}\bar{\tau}_{i}e_{i}-\bar{u}e_{n+1},\underbrace{\sum_{i}\bar{\omega}_{i}e_{i},\ldots,\sum_{i}\bar{\omega}_{i}e_{i}}_{n-1})\\\
&=-(-1)^{n}u\sum\delta_{i_{1},\ldots,i_{n}}^{1,\ldots,n}\bar{\omega}_{i_{1}}\wedge\ldots\wedge\bar{\omega}_{i_{n}}+(-1)^{n}\bar{u}\sum\delta_{i_{1},\ldots,i_{n}}^{1,\ldots,n}\omega_{i_{1}}\wedge\bar{\omega}_{i_{2}}\wedge\ldots\wedge\bar{\omega}_{i_{n}}\\\
&=(-1)^{n}n!(-u+\bar{u}P_{1}(a_{ij}))d\bar{\mu},\end{split}$

where $d\bar{\mu}=\bar{\omega}_{1}\wedge\ldots\wedge\bar{\omega}_{n}$. Since
$M$ and $\bar{M}$ are closed and strictly convex, we can identify $M,\bar{M}$
with $\mathbb{S}^{n}$ by use of Gauss maps. Then previously mentioned
functions and differential forms are now defined on $\mathbb{S}^{n}$.
Therefore, integrating (2.9) over $\mathbb{S}^{n}$ yields

(2.10)
$\displaystyle\int_{\mathbb{S}^{n}}(\bar{u}P_{1}(a_{ij})-u)d\bar{\mu}=0.$

Similarly, we consider a family of differential forms as follows:

(2.11)
$\displaystyle\Phi_{k}=(x,\bar{x},\underbrace{dx,\ldots,dx}_{k},\underbrace{d\bar{x},\ldots,d\bar{x}}_{n-k-1}),\quad
k=1,\ldots,n-1.$

Applying exterior differentiation gives

(2.12) $\displaystyle
d\Phi_{k}=(-1)^{n}n!(\bar{u}P_{k+1}(a_{ij})-uP_{k}(a_{ij}))d\bar{\mu}.$

Then integrating (2.12) over $\mathbb{S}^{n}$ yields

(2.13)
$\displaystyle\int_{\mathbb{S}^{n}}(\bar{u}P_{k+1}(a_{ij})-uP_{k}(a_{ij}))d\bar{\mu}=0.$

In conclusion, we obtain the following integral formulas:

######  Lemma 2.9 ([Ch59]).

Let $x:M^{n}\to\mathbb{R}^{n+1}$ and $\bar{x}:\bar{M}^{n}\to\mathbb{R}^{n+1}$
be two closed smooth strictly convex hypersurfaces. Then, for $k=0,\ldots,n-1$

(2.14)
$\displaystyle\int_{\mathbb{S}^{n}}(\bar{u}P_{k+1}(a_{ij})-uP_{k}(a_{ij}))d\bar{\mu}=0,$
(2.15)
$\displaystyle\int_{\mathbb{S}^{n}}(uP_{k+1}(a^{ij})-\bar{u}P_{k}(a^{ij}))d\mu=0,$

where $a_{ij}=\sum\limits_{s}h^{is}\bar{h}_{sj}$ and
$a^{ij}=\sum\limits_{s}\bar{h}^{is}h_{sj}$.

In particular, when $\bar{M}$ is the unit sphere $\mathbb{S}^{n}$, we have
$\bar{u}=1$ and $(a^{ij})=(h_{ij})$. Note that the eigenvalues of $(h_{ij})$
are the principal curvatures $\\{\kappa_{i}\\}$ of $M$, then $P_{k}(a^{ij})$
is the normalized $k$-th mean curvature $P_{k}(\kappa)$ of $M$. Hence, (2.15)
reduces to Minkowski formulas:

(2.16) $\displaystyle\int_{M}(uP_{k+1}(\kappa)-P_{k}(\kappa))d\mu=0,\quad
k=0,\ldots,n-1.$

Next, we consider the following differential forms:

(2.17)
$\displaystyle\Phi_{k,l}=(x,\bar{x},\underbrace{dx,\ldots,dx}_{k},\underbrace{de_{n+1},\ldots,de_{n+1}}_{n-k-l-1},\underbrace{d\bar{x},\ldots,d\bar{x}}_{l}),\quad
0\leq k+l\leq n-1.$

Denote by $d\sigma$ the standard area element of $\mathbb{S}^{n}$, it follows
that $d\sigma=\omega_{1,n+1}\wedge\ldots\wedge\omega_{n,n+1}$, and then

$d\mu=\det(h^{ij})d\sigma,\quad d\bar{\mu}=\det(\bar{h}^{ij})d\sigma.$

Applying exterior differentiation to (2.17) gives

(2.18)
$\begin{split}d\Phi_{k,l}&=-(-1)^{k+l+1}(n-k-l-1)!u\sum\delta_{i_{1},\ldots
i_{k+l+1}}^{j_{1},\ldots j_{k+l+1}}h^{i_{1}j_{1}}\cdots
h^{i_{k}j_{k}}\bar{h}^{i_{k+1}j_{k+1}}\cdots\bar{h}^{i_{k+l+1}j_{k+l+1}}d\sigma\\\
&\quad+(-1)^{k+l+1}(n-k-l-1)!\bar{u}\sum\delta_{i_{1},\ldots
i_{k+l+1}}^{j_{1},\ldots j_{k+l+1}}h^{i_{1}j_{1}}\cdots
h^{i_{k+1}j_{k+1}}\bar{h}^{i_{k+2}j_{k+2}}\cdots\bar{h}^{i_{k+l+1}j_{k+l+1}}d\sigma\\\
&=(-1)^{k+l+1}n!\left(\bar{u}P_{k+1,l}(h^{ij},\bar{h}^{ij})-uP_{k,l+1}(h^{ij},\bar{h}^{ij})\right)d\sigma.\end{split}$

Hence, integrating (2.18) over $\mathbb{S}^{n}$ yields

######  Lemma 2.10 ([Ch59]).

Let $x:M^{n}\to\mathbb{R}^{n+1}$ and $\bar{x}:\bar{M}^{n}\to\mathbb{R}^{n+1}$
be two closed smooth strictly convex hypersurfaces. Then, for $0\leq k+l\leq
n-1$

(2.19)
$\displaystyle\int_{\mathbb{S}^{n}}\left(\bar{u}P_{k+1,l}(h^{ij},\bar{h}^{ij})-uP_{k,l+1}(h^{ij},\bar{h}^{ij})\right)d\sigma=0.$

Generally, by applying the approach in [Gu10], we can generalize integral
formulas (2.19) to the cases where $u$ and $\bar{u}$ are $C^{2}$ functions.

######  Lemma 2.11.

Let $u$ and $\bar{u}$ be two $C^{2}$ functions on $\mathbb{S}^{n}$. Denote
$(h^{ij})=(u_{ij}+u\delta_{ij})$ and
$(\bar{h}^{ij})=(\bar{u}_{ij}+\bar{u}\delta_{ij})$. Then, for $0\leq k+l\leq
n-1$

(2.20)
$\displaystyle\int_{\mathbb{S}^{n}}\left(\bar{u}P_{k,l}(h^{ij},\bar{h}^{ij})-P_{k,l+1}(h^{ij},\bar{h}^{ij})\right)d\sigma=0,$
(2.21)
$\displaystyle\int_{\mathbb{S}^{n}}\left(uP_{k,l}(h^{ij},\bar{h}^{ij})-P_{k+1,l}(h^{ij},\bar{h}^{ij})\right)d\sigma=0,$

and

(2.22)
$\displaystyle\int_{\mathbb{S}^{n}}\left(\bar{u}P_{k+1,l}(h^{ij},\bar{h}^{ij})-uP_{k,l+1}(h^{ij},\bar{h}^{ij})\right)d\sigma=0.$

Finally, we derive some new integral formulas that contain radial functions.

Let $\eta:(0,+\infty)\to(0,+\infty)$ be a $C^{1}$ function. For any
$k=1,\ldots,n-1$, we calculate the exterior differentiation of
$\eta(\bar{r})\Phi_{k}$:

$\displaystyle
d(\eta(\bar{r})\Phi_{k})=\eta(\bar{r})d\Phi_{k}+\frac{\eta^{\prime}(\bar{r})}{\bar{r}}\sum_{i}\bar{\tau}_{i}\bar{\omega}_{i}\wedge\Phi_{k}.$

Using (2.6), (2.7) and (2.11), we have

$\displaystyle\Phi_{k}$
$\displaystyle=(\sum_{i}\tau_{i}e_{i}-ue_{n+1},\sum_{i}\bar{\tau}_{i}e_{i}-\bar{u}e_{n+1},\sum_{i}\bar{\omega}_{i}e_{i},\underbrace{\sum_{i}\omega_{i}e_{i},\ldots,\sum_{i}\omega_{i}e_{i}}_{k},\underbrace{\sum_{i}\bar{\omega}_{i}e_{i},\ldots,\sum_{i}\bar{\omega}_{i}e_{i}}_{n-k-1})$
$\displaystyle=(-1)^{n}\sum\delta_{i_{1},\ldots,i_{n}}^{1,\ldots,n}(-u\bar{\tau}_{i_{1}}+\bar{u}\tau_{i_{1}})\underbrace{\omega_{i_{2}}\wedge\ldots\wedge\omega_{i_{k+1}}}_{k}\wedge\underbrace{\bar{\omega}_{i_{k+2}}\wedge\ldots\wedge\bar{\omega}_{i_{n}}}_{n-k-1}$
$\displaystyle=(-1)^{n}\sum\delta_{i_{1},\ldots,i_{n}}^{1,\ldots,n}(-u\bar{\tau}_{i_{1}}+\bar{u}\tau_{i_{1}})a_{i_{2}j_{2}}\cdots
a_{i_{k+1}j_{k+1}}\underbrace{\bar{\omega}_{j_{2}}\wedge\ldots\wedge\bar{\omega}_{j_{k+1}}}_{k}\underbrace{\bar{\omega}_{i_{k+2}}\wedge\ldots\wedge\bar{\omega}_{i_{n}}}_{n-k-1}$
$\displaystyle=(-1)^{n}\sum\delta_{i_{1},j_{2}\ldots,j_{k+1},i_{k+2},\ldots,i_{n}}^{1,2,\ldots,k+1,k+2,\ldots,n}(\bar{u}\tau_{i_{1}}-u\bar{\tau}_{i_{1}})a_{i_{2}j_{2}}\cdots
a_{i_{k+1}j_{k+1}}\underbrace{\bar{\omega}_{i_{2}}\wedge\ldots\wedge\bar{\omega}_{i_{n}}}_{n-1}.$

Combining this with (2.12), we obtain

(2.23)
$\begin{split}d(\eta(\bar{r})\Phi_{k})&=(-1)^{n}n!\eta(\bar{r})(\bar{u}P_{k+1}(a_{ij})-uP_{k}(a_{ij}))d\bar{\mu}\\\
&\quad+(-1)^{n}\frac{\eta^{\prime}(\bar{r})}{\bar{r}}\sum\delta_{i_{1},j_{2}\ldots,j_{k+1},i_{k+2},\ldots,i_{n}}^{1,2,\ldots,{k+1},k+2,\ldots,n}\bar{\tau}_{i}(\bar{u}\tau_{i_{1}}-u\bar{\tau}_{i_{1}})a_{i_{2}j_{2}}\cdots
a_{i_{k+1}j_{k+1}}\bar{\omega}_{i}\wedge\underbrace{\bar{\omega}_{i_{2}}\wedge\ldots\wedge\bar{\omega}_{i_{n}}}_{n-1}\\\
&=(-1)^{n}n!\eta(\bar{r})(\bar{u}P_{k+1}(a_{ij})-uP_{k}(a_{ij}))d\bar{\mu}\\\
&\quad+(-1)^{n}(n-k-1)!\frac{\eta^{\prime}(\bar{r})}{\bar{r}}\sum\delta_{i_{1},i_{2}\ldots,i_{k+1}}^{i_{1},j_{2},\ldots,j_{k+1}}(\bar{u}\bar{\tau}_{i_{1}}\tau_{i_{1}}-u\bar{\tau}_{i_{1}}^{2})a_{i_{2}j_{2}}\cdots
a_{i_{k+1}j_{k+1}}d\bar{\mu}.\end{split}$

When $k=0$, applying exterior differentiation of $\eta(\bar{r})\Phi_{0}$ gives

(2.24)
$d(\eta(\bar{r})\Phi_{0})=(-1)^{n}n!\eta(\bar{r})(\bar{u}P_{1}(a_{ij})-u)d\bar{\mu}+(-1)^{n}(n-1)!\frac{\eta^{\prime}(\bar{r})}{\bar{r}}\sum_{i}(\bar{u}\bar{\tau}_{i_{1}}\tau_{i_{1}}-u\bar{\tau}_{i_{1}}^{2})d\bar{\mu}.$

Therefore, integrating (2.23) and (2.24) over $\mathbb{S}^{n}$ yields

######  Lemma 2.12.

Let $x:M^{n}\to\mathbb{R}^{n+1}$ and $\bar{x}:\bar{M}^{n}\to\mathbb{R}^{n+1}$
be two closed smooth strictly convex hypersurfaces. Then, for $k=0,\ldots,n-1$

(2.25)
$\begin{split}&\int_{\mathbb{S}^{n}}\eta(\bar{r})(\bar{u}P_{k+1}(a_{ij})-uP_{k}(a_{ij}))d\bar{\mu}\\\
=&\frac{(n-k-1)!}{n!}\int_{\mathbb{S}^{n}}\frac{\eta^{\prime}(\bar{r})}{\bar{r}}\sum\delta_{i_{1},i_{2}\ldots,i_{k+1}}^{i_{1},j_{2},\ldots,j_{k+1}}(u\bar{\tau}_{i_{1}}^{2}-\bar{u}\bar{\tau}_{i_{1}}\tau_{i_{1}})a_{i_{2}j_{2}}\cdots
a_{i_{k+1}j_{k+1}}d\bar{\mu},\end{split}$

and

(2.26)
$\begin{split}&\int_{\mathbb{S}^{n}}\eta(r)(uP_{k+1}(a^{ij})-\bar{u}P_{k}(a^{ij}))d\mu\\\
=&\frac{(n-k-1)!}{n!}\int_{\mathbb{S}^{n}}\frac{\eta^{\prime}(r)}{r}\sum\delta_{i_{1},i_{2}\ldots,i_{k+1}}^{i_{1},j_{2},\ldots,j_{k+1}}(\bar{u}\tau_{i_{1}}^{2}-u\tau_{i_{1}}\bar{\tau}_{i_{1}})a^{i_{2}j_{2}}\cdots
a^{i_{k+1}j_{k+1}}d\mu,\end{split}$

where the terms $a_{i_{2}j_{2}}\cdots a_{i_{k+1}j_{k+1}}$ and
$a^{i_{2}j_{2}}\cdots a^{i_{k+1}j_{k+1}}$ on the right hand sides vanish when
$k=0$.

Moreover, when $\bar{M}$ is the unit sphere $\mathbb{S}^{n}$, then
$\bar{r}=\bar{u}=1$, $\bar{\tau}_{i}=0$ and $(a^{ij})=(h_{ij})$. Using (2.26),
we obtain

######  Lemma 2.13.

Let $x:M^{n}\to\mathbb{R}^{n+1}$ be a closed smooth strictly convex
hypersurface. Then, for $k=0,\ldots,n-1$

(2.27)
$\begin{split}&\int_{\mathbb{S}^{n}}\eta(r)(uP_{k+1}(h_{ij})-P_{k}(h_{ij}))d\mu\\\
=&\frac{(n-k-1)!}{n!}\int_{\mathbb{S}^{n}}\frac{\eta^{\prime}(r)}{r}\sum\delta_{i_{1},i_{2}\ldots,i_{k+1}}^{i_{1},j_{2},\ldots,j_{k+1}}\tau_{i_{1}}^{2}h_{i_{2}j_{2}}\cdots
h_{i_{k+1}j_{k+1}}d\mu,\end{split}$

where the term $h_{i_{2}j_{2}}\cdots h_{i_{k+1}j_{k+1}}$ on the right hand
side vanishes when $k=0$.

Note that when $\eta\equiv 1$, Lemma 2.12 reduces to Lemma 2.9, and Lemma 2.13
reduces to Minkowski formulas (2.16).

###  2.4. Spectral estimate

In this subsection, we will use the same notation as in subsection 2.3. Let
$(\mathbb{S}^{n},g_{0},\nabla)$ denote the unit sphere equipped with its
standard round metric and Levi-Civita connection. For a smooth closed strictly
convex hypersurface $x:M^{n}\to\mathbb{R}^{n+1}$, denote by
$\nu:M^{n}\to\mathbb{S}^{n}$ the Gauss map of $M$. For convenience, we denote
$\sigma_{k}=\sigma_{k}(h^{ij})$ for $1\leq k\leq n$ and
$dV_{n}=u\sigma_{n}d\sigma$.

The following lemma is the local version of the Alexandrov-Fenchel inequality
and can also be regarded as a spectral interpretation of the Brunn-Minkowski
inequality. For more details, refer to [An97, ACGL20, KM22, Mil21, IM23].

######  Lemma 2.14 ([An97, ACGL20]).

Let $f\in C^{2}(\mathbb{S}^{n})$ with
$\int_{\mathbb{S}^{n}}fu\sigma_{n}d\sigma=0$. Then we have

(2.28)
$n\int_{\mathbb{S}^{n}}f^{2}u\sigma_{n}d\sigma\leq\int_{\mathbb{S}^{n}}u^{2}\sigma_{n}^{ij}\nabla_{i}f\nabla_{j}fd\sigma.$

Equality holds if and only if for some vector $v\in\mathbb{R}^{n+1}$ we have

$f(z)=\langle\frac{z}{u(z)},v\rangle,\quad\forall\ z\in\mathbb{S}^{n}.$

It is well-known (see e.g. [Sch14]) that the inverse Gauss map
$X=\nu^{-1}:\mathbb{S}^{n}\to M$ is given by

$\displaystyle X(z)=u(z)z+\nabla u(z),\quad\forall\ z\in\mathbb{S}^{n}.$

By substituting some test functions in (2.28), we obtain a key inequality as
follows:

######  Lemma 2.15.

Assume that $\alpha\in\mathbb{R}$. Then we have

(2.29) $\begin{split}n\int_{\mathbb{S}^{n}}|X|^{2\alpha+2}dV_{n}&\leq
n\frac{|\int_{\mathbb{S}^{n}}|X|^{\alpha}XdV_{n}|^{2}}{\int_{\mathbb{S}^{n}}dV_{n}}+\int_{\mathbb{S}^{n}}|X|^{2\alpha}u(\Delta
u+nu)dV_{n}\\\
&\quad+(\alpha^{2}+2\alpha)\int_{\mathbb{S}^{n}}|X|^{2\alpha-1}u\langle\nabla
u,\nabla|X|\rangle dV_{n}.\end{split}$

###### Proof.

Let $\\{E_{l}\\}_{l=1}^{n+1}$ be an orthonormal basis of $\mathbb{R}^{n+1}$.
Suppose $\\{e_{i}\\}_{i=1}^{n}$ is a local orthonormal frame for
$\mathbb{S}^{n}$ such that
$(u_{ij}+u\delta_{ij})(z_{0})=\lambda_{i}(z_{0})\delta_{ij}$. Inspired by the
construction of Ivaki and Milman [IM23], for $l=1,\ldots,n+1$, we define the
functions $f_{l}:\mathbb{S}^{n}\to\mathbb{R}$

(2.30) $f_{l}(z)=|X(z)|^{\alpha}\langle
X(z),E_{l}\rangle-\frac{\int_{\mathbb{S}^{n}}|X(z)|^{\alpha}\langle
X(z),E_{l}\rangle dV_{n}}{\int_{\mathbb{S}^{n}}dV_{n}}.$

Since $\int_{\mathbb{S}^{n}}f_{l}dV_{n}=0$ for $1\leq l\leq n+1$, by applying
Lemma 2.14 to $f_{l}$ and summing over $l$, we have

$\displaystyle
n\sum_{l}\int_{\mathbb{S}^{n}}f_{l}^{2}dV_{n}=n\left[\int_{\mathbb{S}^{n}}|X|^{2\alpha+2}dV_{n}-\frac{|\int_{\mathbb{S}^{n}}|X|^{\alpha}XdV_{n}|^{2}}{\int_{\mathbb{S}^{n}}dV_{n}}\right]\leq\sum_{l}\int_{\mathbb{S}^{n}}u^{2}\sigma_{n}^{ij}\nabla_{i}f_{l}\nabla_{j}f_{l}d\sigma.$

It follows from that
$\nabla_{i}X=\sum_{j}(u_{ij}+u\delta_{ij})e_{j}=\lambda_{i}e_{i}$ at $z_{0}$.
Then $\langle e_{i},X\rangle=u_{i}$ and $\lambda_{i}\langle
e_{i},X\rangle^{2}=|X|\langle\nabla u,\nabla|X|\rangle$ at $z_{0}$. Using
$\frac{\partial\sigma_{n}}{\partial\lambda_{i}}=\frac{\sigma_{n}}{\lambda_{i}}$
and
$\sum_{i}\frac{\partial\sigma_{n}}{\partial\lambda_{i}}\lambda_{i}^{2}=\sigma_{1}\sigma_{n}$,
we have

$\displaystyle\sum_{l,i,j}\sigma_{n}^{ij}\nabla_{i}f_{l}\nabla_{j}f_{l}$
$\displaystyle=\sum_{l,i}\frac{\partial\sigma_{n}}{\partial\lambda_{i}}(\nabla_{i}(|X|^{\alpha})\langle
X,E_{l}\rangle+|X|^{\alpha}\langle\nabla_{i}X,E_{l}\rangle)^{2}$
$\displaystyle=\sum_{l,i}\frac{\partial\sigma_{n}}{\partial\lambda_{i}}(\alpha|X|^{\alpha-2}\langle\lambda_{i}e_{i},X\rangle\langle
X,E_{l}\rangle+|X|^{\alpha}\langle\lambda_{i}e_{i},E_{l}\rangle)^{2}$
$\displaystyle=\sum_{i}\frac{\partial\sigma_{n}}{\partial\lambda_{i}}\lambda_{i}^{2}(|X|^{2\alpha}+(\alpha^{2}+2\alpha)|X|^{2\alpha-2}\langle
e_{i},X\rangle^{2})$
$\displaystyle=|X|^{2\alpha}\sigma_{1}\sigma_{n}+(\alpha^{2}+2\alpha)|X|^{2\alpha-1}\langle\nabla
u,\nabla|X|\rangle\sigma_{n}.$

Therefore, we obtain

$\displaystyle
n\left[\int_{\mathbb{S}^{n}}|X|^{2\alpha+2}dV_{n}-\frac{|\int_{\mathbb{S}^{n}}|X|^{\alpha}XdV_{n}|^{2}}{\int_{\mathbb{S}^{n}}dV_{n}}\right]$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{S}^{n}}u^{2}\left(|X|^{2\alpha}\sigma_{1}+(\alpha^{2}+2\alpha)|X|^{2\alpha-1}\langle\nabla
u,\nabla|X|\rangle\right)\sigma_{n}d\sigma$ $\displaystyle=$
$\displaystyle\int_{\mathbb{S}^{n}}u\left(|X|^{2\alpha}(\Delta
u+nu)+(\alpha^{2}+2\alpha)|X|^{2\alpha-1}\langle\nabla
u,\nabla|X|\rangle\right)dV_{n}.$

This completes the proof of Lemma 2.15. ∎

######  Remark 2.1.

In the case $\alpha=0$, Lemma 2.15 reduces to [IM23, Lemma 3.2] for $k=n$.

##  3\. Proofs of Theorem 1.1, Theorem 1.2 and Theorem 1.4

###### Proof of Theorem 1.1.

Suppose there exist two admissible solutions $M$ and $\bar{M}$ to (1.10), i.e.

(3.1)
$\displaystyle\varphi(u)P_{k}(W)=\varphi(\bar{u})P_{k}(\bar{W})=f\quad\text{on}\
\mathbb{S}^{n},$

where $W=(\nabla^{2}u+ug_{0})$ and $\bar{W}=(\nabla^{2}\bar{u}+\bar{u}g_{0}).$

Denote $P_{k,l}=P_{k,l}(W,\bar{W})$. Since $W,\bar{W}\in\Gamma_{k}$, it
follows from (3.1) and Lemma 2.6 that

$\displaystyle P_{k-1,1}\geq
P_{k0}^{\frac{k-1}{k}}P_{0k}^{\frac{1}{k}}=\left(\frac{\varphi(\bar{u})}{\varphi(u)}\right)^{\frac{k-1}{k}}P_{0k},\quad
P_{1,k-1}\geq
P_{k0}^{\frac{1}{k}}P_{0k}^{\frac{k-1}{k}}=\left(\frac{\varphi(u)}{\varphi(\bar{u})}\right)^{\frac{k-1}{k}}P_{k0}.$

For $2\leq k\leq n$, by using integral formulas (2.22) and the assumption
(1.11), we get

(3.2) $\displaystyle 0$
$\displaystyle=\int_{\mathbb{S}^{n}}u(P_{0k}-P_{k-1,1})d\sigma+\int_{\mathbb{S}^{n}}\bar{u}(P_{k0}-P_{1,k-1})d\sigma$
$\displaystyle\leq\int_{\mathbb{S}^{n}}\left[u\left(1-\left(\frac{\varphi(\bar{u})}{\varphi(u)}\right)^{\frac{k-1}{k}}\right)P_{0k}+\bar{u}\left(1-\left(\frac{\varphi(u)}{\varphi(\bar{u})}\right)^{\frac{k-1}{k}}\right)P_{k0}\right]d\sigma$
$\displaystyle=\int_{\mathbb{S}^{n}}\left(1-\left(\frac{\varphi(\bar{u})}{\varphi(u)}\right)^{\frac{k-1}{k}}\right)\left(1-\frac{\bar{u}}{u}\left(\frac{\varphi(\bar{u})}{\varphi(u)}\right)^{\frac{1}{k}}\right)uP_{0k}d\sigma$
$\displaystyle\leq 0.$

Hence the inequalities in (3.2) are actually equalities. It follows that
$W,\bar{W}$ are pairwise proportional and $u=\bar{u}$. Therefore, we obtain
$M=\bar{M}$. ∎

When $\varphi$ is $C^{1}$-smooth, we can reduce the condition (1.11) to a
characterization of $(\log\varphi)^{\prime}$, and then obtain the following
corollary.

######  Corollary 3.1.

Let $2\leq k\leq n$ be an integer. Let $\varphi:(0,\infty)\to(0,\infty)$ be a
$C^{1}$ function that satisfies

(3.3) $\begin{split}&-\frac{k}{s}<(\log\varphi)^{\prime}(s)<0,\end{split}$

for any $s>0$. For any positive function $f\in C(\mathbb{S}^{n})$, then the
smooth admissible solution to (1.10) is unique.

######  Remark 3.1.

Applying the maximum principle, it is clear that the conclusion of Corollary
3.1 still holds when the condition (3.3) is replaced by the condition
$(\log\varphi)^{\prime}(s)<-\frac{k}{s}$.

###### Proof of Theorem 1.4.

Suppose there exist two smooth strictly convex solutions $M$ and $\bar{M}$ to
(1.19). Denote $a_{ij}=\sum_{k}h^{ik}\bar{h}_{kj}$ and
$P_{k,n-k}=P_{k,n-k}(W,\bar{W})$, then we have

(3.4)
$\displaystyle\det(a_{ij})=\frac{P_{n,0}}{P_{0,n}}=\frac{\bar{u}^{1-p}\bar{r}^{q-n-1}}{u^{1-p}r^{q-n-1}}.$

Using the integral formulas (2.25) and (2.26) with $k=0$, we obtain

$\displaystyle\int_{\mathbb{S}^{n}}\left[\eta(\bar{r})(\bar{u}P_{1,n-1}-uP_{0,n})+\eta(r)(uP_{n-1,1}-\bar{u}P_{n,0})\right]d\sigma$
$\displaystyle=$
$\displaystyle\frac{1}{n}\int_{\mathbb{S}^{n}}\left[\frac{\eta^{\prime}(\bar{r})}{\bar{r}}\sum_{i}(u\bar{\tau}_{i}^{2}-\bar{u}\bar{\tau}_{i}\tau_{i})P_{0,n}+\frac{\eta^{\prime}(r)}{r}\sum_{i}(\bar{u}\tau_{i}^{2}-u\tau_{i}\bar{\tau}_{i})P_{n,0}\right]d\sigma$
$\displaystyle\leq$
$\displaystyle\frac{1}{n}\int_{\mathbb{S}^{n}}\left[\frac{\eta^{\prime}(\bar{r})}{\bar{r}}\left(u(\bar{r}^{2}-\bar{u}^{2})-\bar{u}\sqrt{\bar{r}^{2}-\bar{u}^{2}}\sqrt{r^{2}-u^{2}}\right)P_{0,n}\right.$
$\displaystyle\quad\left.+\frac{\eta^{\prime}(r)}{r}\left(\bar{u}(r^{2}-u^{2})-u\sqrt{r^{2}-u^{2}}\sqrt{\bar{r}^{2}-\bar{u}^{2}}\right)P_{n,0}\right]d\sigma$
$\displaystyle=$
$\displaystyle\frac{1}{n}\int_{\mathbb{S}^{n}}\left(\sqrt{\left(\frac{\bar{r}}{\bar{u}}\right)^{2}-1}-\sqrt{\left(\frac{r}{u}\right)^{2}-1}\right)\left(\frac{\eta^{\prime}(\bar{r})}{\bar{r}}\sqrt{\bar{r}^{2}-\bar{u}^{2}}P_{0,n}-\frac{\eta^{\prime}(r)}{r}\sqrt{r^{2}-u^{2}}P_{n,0}\right)u\bar{u}d\sigma.$

where we used the Cauchy-Schwarz inequality. It follows from Lemma 2.6 that

$\displaystyle\int_{\mathbb{S}^{n}}\left[\eta(\bar{r})(\bar{u}P_{1,n-1}-uP_{0,n})+\eta(r)(uP_{n-1,1}-\bar{u}P_{n,0})\right]d\sigma$
$\displaystyle\geq$
$\displaystyle\int_{\mathbb{S}^{n}}\left[\eta(\bar{r})(\bar{u}P_{0,n}^{\frac{n-1}{n}}P_{n,0}^{\frac{1}{n}}-uP_{0,n})+\eta(r)(uP_{n,0}^{\frac{n-1}{n}}P_{0,n}^{\frac{1}{n}}-\bar{u}P_{n,0})\right]d\sigma$
$\displaystyle=$
$\displaystyle\int_{\mathbb{S}^{n}}\left(\frac{\bar{u}}{u}\left(\frac{P_{n,0}}{P_{0,n}}\right)^{\frac{1}{n}}-1\right)\left(1-\frac{\eta(r)}{\eta(\bar{r})}\left(\frac{P_{n,0}}{P_{0,n}}\right)^{\frac{n-1}{n}}\right)u\eta(\bar{r})P_{0,n}d\sigma$

Let $\eta(t)=t^{\alpha}$, where $\alpha$ is a non-positive constant to be
determined. By (3.4), we obtain

(3.5)
$\begin{split}&\int_{\mathbb{S}^{n}}\left(\left(\frac{\bar{u}}{u}\right)^{\frac{n+1-p}{n}}\left(\frac{\bar{r}}{r}\right)^{\frac{q-n-1}{n}}-1\right)\left(1-\left(\frac{\bar{u}}{u}\right)^{\frac{(n-1)(1-p)}{n}}\left(\frac{\bar{r}}{r}\right)^{\frac{(n-1)(q-n-1)-n\alpha}{n}}\right)u\bar{r}^{\alpha}P_{0,n}d\sigma\\\
\leq&\frac{\alpha}{n}\int_{\mathbb{S}^{n}}\left(\bar{u}^{p}\bar{r}^{\alpha+n-1-q}\sqrt{\left(\frac{\bar{r}}{\bar{u}}\right)^{2}-1}-u^{p}r^{\alpha+n-1-q}\sqrt{\left(\frac{r}{u}\right)^{2}-1}\right)\\\
&\quad\quad\left(\sqrt{\left(\frac{\bar{r}}{\bar{u}}\right)^{2}-1}-\sqrt{\left(\frac{r}{u}\right)^{2}-1}\right)u\bar{u}^{2-p}\bar{r}^{q-n-1}P_{0,n}d\sigma.\end{split}$

When $1\leq p<n+1,\ q\leq n+1$, we choose
$\alpha=-\frac{(n-1)(n+1-q)}{n+1-p}\leq 0$. Let $\rho=\frac{\bar{u}}{u}$ and
$\bar{\rho}=\frac{\bar{r}}{r}$, we have for any $\rho,\bar{\rho}>0$

$\displaystyle\left(\rho^{\frac{n+1-p}{n}}\bar{\rho}^{\frac{q-n-1}{n}}-1\right)\left(1-\rho^{\frac{(n-1)(1-p)}{n}}\bar{\rho}^{\frac{(n-1)(q-n-1)-n\alpha}{n}}\right)\geq
0.$

When $p=2$, then $\alpha=q-n-1$. Let $\xi=\frac{r}{u}$ and
$\bar{\xi}=\frac{\bar{r}}{\bar{u}}$, we have for any
$1\leq\xi,\bar{\xi}\leq\sqrt{2}$

$\displaystyle\left(\bar{\xi}^{-2}\sqrt{\bar{\xi}^{2}-1}-\xi^{-2}\sqrt{\xi^{2}-1}\right)\left(\sqrt{\bar{\xi}^{2}-1}-\sqrt{\xi^{2}-1}\right)\geq
0,$

where we used the monotonic increasing property of $g(t)=t^{-2}\sqrt{t^{2}-1}$
for $1\leq t\leq\sqrt{2}$.

Consequently, the left hand side of (3.5) is non-negative and the right hand
side of (3.5) is non-positive. Then both sides must be zero. It follows that
$W=\lambda\bar{W}$ and $\tau_{i}=\lambda^{*}\bar{\tau}_{i}$, $1\leq i\leq n$,
for some $\lambda,\lambda^{*}>0$. If $p\neq q$, we have $M=\bar{M}$. If $p=q$,
we have $M=\lambda\bar{M}$. ∎

###### Proof of Theorem 1.2.

Without loss of generality, we can assume $\psi(1,1)=1$, then $\psi(s,t)$ and
$\eta(t)$ satisfy

(3.6)
$\displaystyle\left(s-\psi(s,t)\right)\left(\eta(1)-\eta(t)\psi^{k-l-1}(s,t)\right)\leq
0,$

for any $t>s>0$. Suppose there exists a smooth strictly convex solution $M$ to
(1.12). Let $\bar{M}$ be the unit sphere, then $\bar{M}$ is a strictly convex
solution to (1.12).

Using the integral formulas (2.25) and (2.26), we have

$\displaystyle\int_{\mathbb{S}^{n}}\left[\eta(1)(P_{l+1}(W)-uP_{l}(W))+\eta(r)(uP_{k}(W)-P_{k-1}(W))\right]d\sigma$
$\displaystyle=$
$\displaystyle\frac{(k-1)!}{n!}\int_{\mathbb{S}^{n}}\frac{\eta^{\prime}(r)}{r}\sum\delta_{i_{1},i_{2}\ldots,i_{n-k+1}}^{i_{1},j_{2},\ldots,j_{n-k+1}}\tau_{i_{1}}^{2}h_{i_{2}j_{2}}\cdots
h_{i_{n-k+1}j_{n-k+1}}P_{n}(W)d\sigma.$

Note that for a fixed $z_{0}\in\mathbb{S}^{n}$, after choosing a local
orthogonal frame such that $h_{ij}=\kappa_{i}\delta_{ij}$ at $z_{0}$, then we
have at $z_{0}$

$\displaystyle\sum\delta_{i_{1},i_{2}\ldots,i_{n-k+1}}^{i_{1},j_{2},\ldots,j_{n-k+1}}\tau_{i_{1}}^{2}h_{i_{2}j_{2}}\cdots
h_{i_{n-k+1}j_{n-k+1}}$
$\displaystyle=\sum\tau_{i_{1}}^{2}\kappa_{i_{2}}\cdots\kappa_{i_{n-k+1}}\geq
0.$

Then it follows from $\eta^{\prime}(t)\leq 0$ and Lemma 2.3 that

$\displaystyle 0$
$\displaystyle\geq\int_{\mathbb{S}^{n}}\left[\eta(1)P_{l}(W)\left(\frac{P_{l+1}(W)}{P_{l}(W)}-u\right)+\eta(r)P_{k}(W)\left(u\frac{P_{k-1}(W)}{P_{k}(W)}-1\right)\right]d\sigma$
$\displaystyle\geq\int_{\mathbb{S}^{n}}\left[\eta(1)P_{l}(W)\left(\left(\frac{P_{k}(W)}{P_{l}(W)}\right)^{\frac{1}{k-l}}-u\right)+\eta(r)P_{k}(W)\left(u\left(\frac{P_{k}(W)}{P_{l}(W)}\right)^{-\frac{1}{k-l}}-1\right)\right]d\sigma$
$\displaystyle=\int_{\mathbb{S}^{n}}\left(u-\left(\frac{P_{k}(W)}{P_{l}(W)}\right)^{\frac{1}{k-l}}\right)\left(\eta(r)\left(\frac{P_{k}(W)}{P_{l}(W)}\right)^{\frac{k-l-1}{k-l}}-\eta(1)\right)P_{l}(W)d\sigma.$

Using (1.12) and the assumption (3.6), we obtain

(3.7)
$\begin{split}0&\geq\int_{\mathbb{S}^{n}}\left(u-\psi(u,r)\right)\left(\eta(r)\psi^{k-l-1}(u,r)-\eta(1)\right)P_{l}(W)d\sigma\geq
0.\end{split}$

Hence the inequalities in (3.7) are both equalities. It follows that
$W=\lambda I$, and then $M$ is a sphere. Moreover, if in addition
$\eta^{\prime}(t)<0$, then we have $\tau_{i}\equiv 0,\ 1\leq i\leq n$. Thus
$r=u$ and $M$ is an origin-centred sphere. ∎

##  4\. Proof of Theorem 1.3

###### Proof of Theorem 1.3.

Suppose there exists a smooth strictly convex solution $M$ to (1.12). Let
$\\{e_{1},\ldots,e_{n},e_{n+1}\\}$ be a local orthonormal frame at $x\in M$
such that $e_{n+1}$ is the unit inner normal vector of $M$ and
$h_{ij}=\kappa_{i}\delta_{ij}$ at $x$, then we have $u_{i}=\kappa_{i}\langle
x,e_{i}\rangle$ and $r_{i}=r^{-1}\langle x,e_{i}\rangle$ at $x$.

Denote $\tilde{k}=n-k,\ \tilde{l}=n-l$, $\tilde{\psi}=\psi^{-1}$ and
$P_{i}=P_{i}(\kappa)$ for $1\leq i\leq n$, we have
$0\leq\tilde{k}<\tilde{l}\leq n$, $\partial_{1}\tilde{\psi}\leq 0$ and
$\partial_{2}\tilde{\psi}\leq 0$. Note that
$P_{i}(W)=P_{i}(\kappa^{-1})=\frac{P_{n-i}(\kappa)}{P_{n}(\kappa)}$, then
(1.13) is equivalent to

(4.1)
$\left(\frac{P_{\tilde{l}}}{P_{\tilde{k}}}\right)^{\frac{1}{\tilde{l}-\tilde{k}}}=\tilde{\psi}(u,r).$

When $\tilde{k}<\tilde{l}-1$, by using Minkowski formulas (2.16), Lemma 2.3,
Proposition 2.8 and (4.1), we obtain

(4.2)
$\begin{split}0&\leq\int_{M}(P_{\tilde{l}-1}-P_{\tilde{k}}^{\frac{1}{\tilde{l}-\tilde{k}}}P_{\tilde{l}}^{\frac{\tilde{l}-\tilde{k}-1}{\tilde{l}-\tilde{k}}})d\mu+\int_{M}u(P_{\tilde{k}+1}-P_{\tilde{k}}^{\frac{\tilde{l}-\tilde{k}-1}{\tilde{l}-\tilde{k}}}P_{\tilde{l}}^{\frac{1}{\tilde{l}-\tilde{k}}})\left(\frac{P_{\tilde{l}}}{P_{\tilde{k}}}\right)^{\frac{\tilde{l}-\tilde{k}-1}{\tilde{l}-\tilde{k}}}d\mu\\\
&=\int_{M}(P_{\tilde{l}-1}-uP_{\tilde{l}})d\mu+\int_{M}(uP_{\tilde{k}+1}-P_{\tilde{k}})\left(\frac{P_{\tilde{l}}}{P_{\tilde{k}}}\right)^{\frac{\tilde{l}-\tilde{k}-1}{\tilde{l}-\tilde{k}}}d\mu\\\
&\leq\int_{M}(uP_{\tilde{k}+1}-P_{\tilde{k}})\left(\frac{P_{\tilde{l}}}{P_{\tilde{k}}}\right)^{\frac{\tilde{l}-\tilde{k}-1}{\tilde{l}-\tilde{k}}}d\mu=\frac{1}{(n-\tilde{k})\binom{n}{\tilde{k}}}\int_{M}T_{ij}^{\tilde{k}}\langle
x,e_{i}\rangle\left(\left(\frac{P_{\tilde{l}}}{P_{\tilde{k}}}\right)^{\frac{\tilde{l}-\tilde{k}-1}{\tilde{l}-\tilde{k}}}\right)_{j}d\mu\\\
&=\frac{\tilde{l}-\tilde{k}-1}{(n-\tilde{k})\binom{n}{\tilde{k}}}\int_{M}T_{ij}^{\tilde{k}}\langle
x,e_{i}\rangle(\tilde{\psi}(u,r))_{j}\left(\frac{P_{\tilde{l}}}{P_{\tilde{k}}}\right)^{\frac{\tilde{l}-\tilde{k}-2}{\tilde{l}-\tilde{k}}}d\mu\\\
&=\frac{\tilde{l}-\tilde{k}-1}{(n-\tilde{k})\binom{n}{\tilde{k}}}\int_{M}T_{ij}^{\tilde{k}}\langle
x,e_{i}\rangle(\kappa_{j}\langle
x,e_{j}\rangle\partial_{1}\tilde{\psi}+r^{-1}\langle
x,e_{j}\rangle\partial_{2}\tilde{\psi})\left(\frac{P_{\tilde{l}}}{P_{\tilde{k}}}\right)^{\frac{\tilde{l}-\tilde{k}-2}{\tilde{l}-\tilde{k}}}d\mu\\\
&\leq 0.\end{split}$

When $\tilde{k}=\tilde{l}-1\geq 1$, we obtain

(4.3)
$\begin{split}0&\leq\int_{M}(P_{\tilde{k}}^{2}-P_{\tilde{k}-1}P_{\tilde{k}+1})\frac{1}{P_{\tilde{k}}}d\mu=\int_{M}(P_{\tilde{k}}-uP_{\tilde{k+1}})d\mu+\int_{M}(uP_{\tilde{k}}-P_{\tilde{k}-1})\frac{P_{\tilde{k}+1}}{P_{\tilde{k}}}d\mu\\\
&=\int_{M}(uP_{\tilde{k}}-P_{\tilde{k}-1})\frac{P_{\tilde{k}+1}}{P_{\tilde{k}}}d\mu=\frac{1}{(n-\tilde{k}+1)\binom{n}{\tilde{k}-1}}\int_{M}T_{ij}^{\tilde{k}-1}\langle
x,e_{i}\rangle\left(\frac{P_{\tilde{k}+1}}{P_{\tilde{k}}}\right)_{j}d\mu\\\
&=\frac{1}{(n-\tilde{k}+1)\binom{n}{\tilde{k}-1}}\int_{M}T_{ij}^{\tilde{k}-1}\langle
x,e_{i}\rangle(\kappa_{j}\langle
x,e_{j}\rangle\partial_{1}\tilde{\psi}+r^{-1}\langle
x,e_{j}\rangle\partial_{2}\tilde{\psi})d\mu\\\ &\leq 0.\end{split}$

When $\tilde{k}=\tilde{l}-1=0$, since $n\geq 2$, we obtain

(4.4)
$\begin{split}0&\leq\int_{M}u(P_{1}^{2}-P_{2})d\mu=\int_{M}(P_{1}-uP_{2})d\mu+\int_{M}(uP_{1}-1)P_{1}d\mu\\\
&=\int_{M}(uP_{1}-1)P_{1}d\mu=\frac{1}{n}\int_{M}\delta_{ij}\langle
x,e_{i}\rangle\left(P_{1}\right)_{j}d\mu\\\
&=\frac{1}{n}\int_{M}\sum_{i}\langle x,e_{i}\rangle(\kappa_{i}\langle
x,e_{i}\rangle\partial_{1}\tilde{\psi}+r^{-1}\langle
x,e_{i}\rangle\partial_{2}\tilde{\psi})d\mu\\\ &\leq 0.\end{split}$

Therefore, the above inequalities in (4.2), (4.3) and (4.4) are all
equalities. It follows that $W=\lambda I$, $\langle x,e_{i}\rangle\equiv 0\
(1\leq i\leq n)$, and then $M$ is an origin-centred sphere. ∎

Furthermore, we study the following equation

(4.5)
$\sum\limits_{s=1}^{n-k}f_{s}(u,r)\frac{P_{k+s}(\kappa)}{P_{k}(\kappa)}+\sum\limits_{s=1}^{n-k}g_{s}(u,r)\frac{P_{k-1+s}(\kappa)}{P_{k-1}(\kappa)}=\psi(u.r),$

where $\kappa$ are the principal curvatures of the hypersurface. We can prove
the following uniqueness result, which generalizes the result in [Si67].

######  Proposition 4.1.

Let $n\geq 2$ and $0<k<n$. Suppose
$\psi:(0,\infty)\times(0,\infty)\to(0,\infty)$ is a $C^{1}$ function with
$\partial_{1}\psi\leq 0,\ \partial_{2}\psi\leq 0$ and at least one of these
inequalities is strict. For $1\leq s\leq n-k$, suppose $f_{s},\
g_{s}:(0,\infty)\times(0,\infty)\to(0,\infty)$ are $C^{1}$ functions with
$f_{s}+g_{s}=\phi_{s}$ and $\partial_{1}\phi_{s}\geq 0,\
\partial_{2}\phi_{s}\geq 0$. Then the smooth strictly convex solution $M$ to
(4.5) must be an origin-centred sphere.

###### Proof of Proposition 4.1.

Suppose there exists a smooth strictly convex solution $M$ to (4.5). Let
$\\{e_{1},\ldots,e_{n},e_{n+1}\\}$ be a local orthonormal frame at $x\in M$
such that $e_{n+1}$ is the unit inner normal vector of $M$ and
$h_{ij}=\kappa_{i}\delta_{ij}$ at $x$. Denote $P_{i}=P_{i}(\kappa)$.

Using Minkowski formulas (2.16), Lemma 2.3, Proposition 2.8 and (4.5), we
obtain

(4.6)
$\begin{split}0&\leq\sum\limits_{s=1}^{n-k}\int_{M}(P_{k}P_{k-1+s}-P_{k-1}P_{k+s})\left(\frac{f_{s}}{P_{k}}+u\frac{g_{s}}{P_{k-1}}\right)d\mu\\\
&=\sum\limits_{s=1}^{n-k}\int_{M}(P_{k-1+s}-uP_{k+s})(f_{s}+g_{s})d\mu-\sum\limits_{s=1}^{n-k}\int_{M}(P_{k-1}-uP_{k})\left(\frac{f_{s}P_{k+s}}{P_{k}}+u\frac{g_{s}P_{k-1+s}}{P_{k-1}}\right)d\mu\\\
&=\sum\limits_{s=1}^{n-k}\int_{M}(P_{k-1+s}-uP_{k+s})\phi_{s}d\mu-\int_{M}(P_{k-1}-uP_{k})\psi
d\mu\\\ &=-\sum\limits_{s=1}^{n-k}\frac{\int_{M}T_{ij}^{k-1+s}\langle
x,e_{i}\rangle(\phi_{s})_{j}d\mu}{(n-k+1-s)\binom{n}{k-1+s}}+\frac{\int_{M}T_{ij}^{k-1}\langle
x,e_{i}\rangle(\psi)_{j}d\mu}{(n-k+1)\binom{n}{k-1}}\\\ &\leq 0.\end{split}$

Hence the above inequalities in (4.6) are both equalities. It follows that
$W=\lambda I$, $\langle x,e_{i}\rangle\equiv 0\ (1\leq i\leq n)$, and then $M$
is an origin-centred sphere. ∎

##  5\. Proofs of Theorem 1.5 and Theorem 1.6

###### Proof of Theorem 1.5.

Suppose there exists a smooth strictly origin-symmetric convex solution $M$ to
(1.8). Then we have $dV_{n}=u^{p}|X|^{n+1-q}d\sigma$.

Since $M$ is origin-symmetric, then
$\int_{\mathbb{S}^{n}}|X|^{\alpha}XdV_{n}=0$ for any $\alpha\in\mathbb{R}$.
Using (2.29), we have

(5.1)
$n\int_{\mathbb{S}^{n}}|X|^{2\alpha+2}dV_{n}\leq\int_{\mathbb{S}^{n}}|X|^{2\alpha}u(\Delta
u+nu)dV_{n}+(\alpha^{2}+2\alpha)\int_{\mathbb{S}^{n}}|X|^{2\alpha-1}u\langle\nabla
u,\nabla|X|\rangle dV_{n}.$

Note that $|X|^{2}=u^{2}+|\nabla u|^{2}$ and

(5.2) $\displaystyle\langle\nabla
u,\nabla|X|^{2}\rangle=\sum_{j}2u_{j}(uu_{j}+\sum_{i}u_{i}u_{ij})=\sum_{i,j}2u_{i}u_{j}(u_{ij}+u\delta_{ij})\geq
c|\nabla u|^{2},$

where $c>0$ depends on $M$. By integration by parts, we have

(5.3) $\begin{split}&\int_{\mathbb{S}^{n}}|X|^{2\alpha}u\Delta
udV_{n}=\int_{\mathbb{S}^{n}}|X|^{2\alpha+n+1-q}u^{p+1}\Delta ud\sigma\\\
=&-(p+1)\int_{\mathbb{S}^{n}}|X|^{2\alpha+n+1-q}u^{p}|\nabla
u|^{2}d\sigma-(2\alpha+n+1-q)\int_{\mathbb{S}^{n}}|X|^{2\alpha+n-q}u^{p+1}\langle\nabla
u,\nabla|X|\rangle d\sigma\\\
=&-(p+1)\int_{\mathbb{S}^{n}}|X|^{2\alpha}|\nabla
u|^{2}dV_{n}-(2\alpha+n+1-q)\int_{\mathbb{S}^{n}}|X|^{2\alpha-1}u\langle\nabla
u,\nabla|X|\rangle d\sigma.\end{split}$

Substituting (5.3) into (5.1), we have

(5.4) $(n+p+1)\int_{\mathbb{S}^{n}}|X|^{2\alpha}|\nabla
u|^{2}dV_{n}\leq(\alpha^{2}+q-n-1)\int_{\mathbb{S}^{n}}|X|^{2\alpha-1}u\langle\nabla
u,\nabla|X|\rangle dV_{n}.$

When $p\geq-n-1$ and $q\leq n+1$ with at least one of these being strict,
choosing $\alpha=0$, it follows from (5.2) and (5.4) that $\nabla u\equiv 0$
on $\mathbb{S}^{n}$. Hence $M$ is an origin-centred sphere.

When $p>-n-1$ and $q>n+1$, using the same notation as in the proof of Lemma
2.15, we have

(5.5) $\begin{split}&\int_{\mathbb{S}^{n}}|X|^{2\alpha-1}u\langle\nabla
u,\nabla|X|\rangle
dV_{n}=\int_{\mathbb{S}^{n}}|X|^{2\alpha-2}u\sum_{i}\lambda_{i}u_{i}^{2}dV_{n}\\\
\leq&\int_{\mathbb{S}^{n}}|X|^{2\alpha-2}u|\nabla u|^{2}(\Delta
u+nu)dV_{n}.\end{split}$

Assume that $\alpha+\frac{n+1-q}{2}\geq 0$, by (5.2), we calculate

(5.6) $\begin{split}&\int_{\mathbb{S}^{n}}|X|^{2\alpha-2}u|\nabla u|^{2}\Delta
udV_{n}=\int_{\mathbb{S}^{n}}|X|^{2\alpha+n-1-q}u^{p+1}|\nabla u|^{2}\Delta
ud\sigma\\\ =&-(p+1)\int_{\mathbb{S}^{n}}|X|^{2\alpha+n-1-q}u^{p}|\nabla
u|^{4}d\sigma-\int_{\mathbb{S}^{n}}|X|^{2\alpha+n-1-q}u^{p+1}\langle\nabla
u,\nabla|\nabla u|^{2}\rangle d\sigma\\\
&\quad-\left(\alpha+\frac{n-1-q}{2}\right)\int_{\mathbb{S}^{n}}|X|^{2\alpha+n-3-q}u^{p+1}|\nabla
u|^{2}\langle\nabla u,\nabla|X|^{2}\rangle d\sigma\\\
=&-(p+1)\int_{\mathbb{S}^{n}}|X|^{2\alpha+n-1-q}u^{p}|\nabla
u|^{4}d\sigma+2\int_{\mathbb{S}^{n}}|X|^{2\alpha+n-1-q}u^{p+2}|\nabla
u|^{2}d\sigma\\\
&\quad-\int_{\mathbb{S}^{n}}\left(|X|^{2}+\left(\alpha+\frac{n-1-q}{2}\right)|\nabla
u|^{2}\right)|X|^{2\alpha+n-3-q}u^{p+1}\langle\nabla u,\nabla|X|^{2}\rangle
d\sigma\\\ \leq&-(p+1)\int_{\mathbb{S}^{n}}|X|^{2\alpha+n-1-q}u^{p}|\nabla
u|^{4}d\sigma+2\int_{\mathbb{S}^{n}}|X|^{2\alpha+n-1-q}u^{p+2}|\nabla
u|^{2}d\sigma\\\ =&-(p+1)\int_{\mathbb{S}^{n}}|X|^{2\alpha-2}|\nabla
u|^{4}dV_{n}+2\int_{\mathbb{S}^{n}}|X|^{2\alpha-2}u^{2}|\nabla
u|^{2}dV_{n}.\end{split}$

It follows from (5.4), (5.5) and (5.6) that

(5.7)
$\begin{split}0&\leq\left(-(n+p+1)-(p+1)(\alpha^{2}+q-n-1)\right)\int_{\mathbb{S}^{n}}|X|^{2\alpha-2}|\nabla
u|^{4}dV_{n}\\\
&\quad+\left(-(n+p+1)+(n+2)(\alpha^{2}+q-n-1)\right)\int_{\mathbb{S}^{n}}|X|^{2\alpha-2}u^{2}|\nabla
u|^{2}dV_{n}.\end{split}$

Notice that $-(p+1)<n+2$. We choose $\alpha=\sqrt{n+1-q+\frac{n+p+1}{n+2}}$
according to the provided conditions for $p$ and $q$. This choice ensures that
$\alpha+\frac{n+1-q}{2}\geq 0$. Consequently, (5.7) becomes

(5.8)
$0\leq-\frac{(n+p+3)(n+p+1)}{n+2}\int_{\mathbb{S}^{n}}|X|^{2\alpha-2}|\nabla
u|^{4}dV_{n}\leq 0.$

Hence $\nabla u\equiv 0$ on $\mathbb{S}^{n}$. The proof of Case (i) is
complete.

Moreover, using the duality relation of $L_{p}$ dual Minkowski problems in
[CHZ19, Theorem 7.1], we have that the solutions to (1.8) with parameters
$(p,q)$ are in one-to-one correspondence with the solutions to (1.8) with
parameters $(-q,-p)$. Combining this with Case (i), we deduce the proof of
Case (ii). We complete the proof of Theorem 1.5. ∎

###### Proof of Theorem 1.6.

Suppose there exists a smooth strictly convex solution $M$ to (1.8). Let
$\alpha$ be a constant to be determined such that $\alpha+\frac{n+1-q}{2}\geq
0$. By (2.29), we have

(5.9) $\begin{split}n\int_{\mathbb{S}^{n}}|X|^{2\alpha+2}dV_{n}&\leq
n\frac{|\int_{\mathbb{S}^{n}}|X|^{\alpha}XdV_{n}|^{2}}{\int_{\mathbb{S}^{n}}dV_{n}}+\int_{\mathbb{S}^{n}}|X|^{2\alpha}u(\Delta
u+nu)dV_{n}\\\
&\quad+(\alpha^{2}+2\alpha)\int_{\mathbb{S}^{n}}|X|^{2\alpha-1}u\langle\nabla
u,\nabla|X|\rangle dV_{n}.\end{split}$

Using the same procedure as Theorem 1.5, we can obtain (5.3), (5.5) and (5.6)
again. Since $q-n-1\geq 0$, we choose $\alpha=q-n-1$. It follows from the
divergence theorem that

$\begin{split}\int_{\mathbb{S}^{n}}|X|^{\alpha}XdV_{n}&=\int_{\mathbb{S}^{n}}u^{p}Xd\sigma=\int_{\mathbb{S}^{n}}u^{p}\nabla
ud\sigma+\int_{\mathbb{S}^{n}}u^{p+1}zd\sigma(z)\\\
&=\frac{n+p+1}{n}\int_{\mathbb{S}^{n}}u^{p}\nabla
ud\sigma=\frac{n+p+1}{n}\int_{\mathbb{S}^{n}}|X|^{\alpha}\nabla
udV_{n}.\end{split}$

Combining this with Cauchy-Schwarz inequality, we have

(5.10)
$\begin{split}n\frac{|\int_{\mathbb{S}^{n}}|X|^{\alpha}XdV_{n}|^{2}}{\int_{\mathbb{S}^{n}}dV_{n}}\leq&\frac{(n+p+1)^{2}}{n}\frac{(\int_{\mathbb{S}^{n}}|X|^{\alpha}|\nabla
u|dV_{n})^{2}}{\int_{\mathbb{S}^{n}}dV_{n}}\\\
\leq&\frac{(n+p+1)^{2}}{n}\int_{\mathbb{S}^{n}}|X|^{2\alpha}|\nabla
u|^{2}dV_{n}.\end{split}$

Substituting (5.3), (5.5) (5.6) and (5.10) into (5.9), we obtain

(5.11)
$\begin{split}0&\leq\left(\frac{(p+1)(n+p+1)}{n}-(p+1)(q-n-1)(q-n)\right)\int_{\mathbb{S}^{n}}|X|^{2\alpha-2}|\nabla
u|^{4}dV_{n}\\\
&\quad+\left(\frac{(p+1)(n+p+1)}{n}+(n+2)(q-n-1)(q-n)\right)\int_{\mathbb{S}^{n}}|X|^{2\alpha-2}u^{2}|\nabla
u|^{2}dV_{n}.\end{split}$

Due to $(p+1)(n+p+1)\leq 0$ and $n+1\leq q\leq
n+\frac{1}{2}+\sqrt{\frac{1}{4}-\frac{(p+1)(n+p+1)}{n(n+2)}}$, we obtain

(5.12) $\begin{split}&\frac{(p+1)(n+p+1)}{n}-(p+1)(q-n-1)(q-n)\\\ \leq\
&\frac{(p+1)(n+p+1)}{n}+(n+2)(q-n-1)(q-n)\\\ \leq\ &0.\end{split}$

When $-n-1<p<-1$, then at least one of inequalities in (5.12) is strict. It
follows from (5.11) that $\nabla u\equiv 0$ on $\mathbb{S}^{n}$. When $p=-1$,
then $q=n+1$ and $\alpha=0$. In this case, (5.11) becomes an equality. It
follows from the equality case of (5.10) that $\nabla u\equiv 0$ on
$\mathbb{S}^{n}$. The proof of Case (i) is complete. Using the duality
relation in [CHZ19, Theorem 7.1], we deduce the proof of Case (ii) and
complete the proof of Theorem 1.6. ∎

##  Appendix A Uniqueness of the $L_{p}$ dual Christoffel-Minkowski problem

In this appendix, by modifying the test functions in [IM23, Lemma 3.1] for
$k<n$, we provide a uniqueness result of origin-symmetric solutions to the
isotropic $L_{p}$ dual Christoffel-Minkowski problem. Denote
$dV_{k}=u\sigma_{k}d\sigma$. We first present a generalization of Lemma 2.15,
which reduces to [IM23, Lemma 3.2] when $\alpha=0$.

######  Lemma A.1.

Let $1\leq k\leq n$. Assume that $\alpha\geq 0$. Then we have

(A.1) $\begin{split}k\int_{\mathbb{S}^{n}}|X|^{2\alpha+2}dV_{k}&\leq
k\frac{|\int_{\mathbb{S}^{n}}|X|^{\alpha}XdV_{k}|^{2}}{\int_{\mathbb{S}^{n}}dV_{k}}+\int_{\mathbb{S}^{n}}|X|^{2\alpha}u\left(\sigma_{1}-(k+1)\frac{\sigma_{k+1}}{\sigma_{k}}\right)dV_{k}\\\
&\quad+(\alpha^{2}+2\alpha)\int_{\mathbb{S}^{n}}|X|^{2\alpha-2}|\nabla
u|^{2}u\left(\sigma_{1}-(k+1)\frac{\sigma_{k+1}}{\sigma_{k}}\right)dV_{n}.\end{split}$

###### Proof.

Let $\\{E_{l}\\}_{l=1}^{n+1}$ be an orthonormal basis of $\mathbb{R}^{n+1}$.
Suppose $\\{e_{i}\\}_{i=1}^{n}$ is a local orthonormal frame for
$\mathbb{S}^{n}$ such that
$(u_{ij}+u\delta_{ij})(z_{0})=\lambda_{i}(z_{0})\delta_{ij}$. For
$l=1,\ldots,n+1$, we define the functions $f_{l}:\mathbb{S}^{n}\to\mathbb{R}$

(A.2) $f_{l}(z)=|X(z)|^{\alpha}\langle
X(z),E_{l}\rangle-\frac{\int_{\mathbb{S}^{n}}|X(z)|^{\alpha}\langle
X(z),E_{l}\rangle dV_{k}}{\int_{\mathbb{S}^{n}}dV_{k}}.$

Since $\int_{\mathbb{S}^{n}}f_{l}dV_{k}=0$ for $1\leq l\leq n+1$, applying
Lemma 2.14 to $f_{l}$ and summing over $l$, we have

$\displaystyle
k\sum_{l}\int_{\mathbb{S}^{n}}f_{l}^{2}dV_{k}=k\left[\int_{\mathbb{S}^{n}}|X|^{2\alpha+2}dV_{k}-\frac{|\int_{\mathbb{S}^{n}}|X|^{\alpha}XdV_{k}|^{2}}{\int_{\mathbb{S}^{n}}dV_{k}}\right]\leq\sum_{l}\int_{\mathbb{S}^{n}}u^{2}\sigma_{k}^{ij}\nabla_{i}f_{l}\nabla_{j}f_{l}d\sigma.$

Note that $\sum_{i}\langle e_{i},X\rangle^{2}=|\nabla u|^{2}$ and
$\sum_{i}\frac{\partial\sigma_{k}}{\partial\lambda_{i}}\lambda_{i}^{2}=\sigma_{1}\sigma_{k}-(k+1)\sigma_{k+1}$.
It follows from $(u_{ij}+u\delta_{ij})(z_{0})=\lambda_{i}(z_{0})\delta_{ij}$
that $\nabla_{i}X=\sum_{j}(u_{ij}+u\delta_{ij})e_{j}=\lambda_{i}e_{i}$ at
$z_{0}$. Since $\alpha\geq 0$, we have

$\displaystyle\sum_{l,i,j}\sigma_{k}^{ij}\nabla_{i}f_{l}\nabla_{j}f_{l}$
$\displaystyle=\sum_{l,i}\frac{\partial\sigma_{k}}{\partial\lambda_{i}}(\nabla_{i}(|X|^{\alpha})\langle
X,E_{l}\rangle+|X|^{\alpha}\langle\nabla_{i}X,E_{l}\rangle)^{2}$
$\displaystyle=\sum_{l,i}\frac{\partial\sigma_{k}}{\partial\lambda_{i}}(\alpha|X|^{\alpha-2}\langle\lambda_{i}e_{i},X\rangle\langle
X,E_{l}\rangle+|X|^{\alpha}\langle\lambda_{i}e_{i},E_{l}\rangle)^{2}$
$\displaystyle=\sum_{i}\frac{\partial\sigma_{k}}{\partial\lambda_{i}}\lambda_{i}^{2}(|X|^{2\alpha}+(\alpha^{2}+2\alpha)|X|^{2\alpha-2}\langle
e_{i},X\rangle^{2})$
$\displaystyle\leq(|X|^{2\alpha}+(\alpha^{2}+2\alpha)|X|^{2\alpha-2}|\nabla
u|^{2})(\sigma_{1}\sigma_{k}-(k+1)\sigma_{k+1}).$

Therefore, we obtain

$\displaystyle
k\left[\int_{\mathbb{S}^{n}}|X|^{2\alpha+2}dV_{k}-\frac{|\int_{\mathbb{S}^{k}}|X|^{\alpha}XdV_{k}|^{2}}{\int_{\mathbb{S}^{n}}dV_{n}}\right]$
$\displaystyle\leq$
$\displaystyle\int_{\mathbb{S}^{n}}u^{2}(|X|^{2\alpha}+(\alpha^{2}+2\alpha)|X|^{2\alpha-2}|\nabla
u|^{2})(\sigma_{1}\sigma_{k}-(k+1)\sigma_{k+1})d\sigma$ $\displaystyle=$
$\displaystyle\int_{\mathbb{S}^{n}}u(|X|^{2\alpha}+(\alpha^{2}+2\alpha)|X|^{2\alpha-2}|\nabla
u|^{2})\left(\sigma_{1}-(k+1)\frac{\sigma_{k+1}}{\sigma_{k}}\right)dV_{k}.$

This completes the proof of Lemma A.1. ∎

Now we consider the following isotropic $L_{p}$ dual Christoffel-Minkowski
problem

(A.3) $u^{1-p}r^{q-k-1}\sigma_{k}(W)=1\quad\text{on}\ \mathbb{S}^{n}.$

Corollary 1.5 states the uniqueness of solutions to (A.3) when $p\geq 1$ and
$q\leq k+1$ with at least one of these inequalities being strict. As for the
case with the origin-symmetric assumption, in [IM23, Theorem 1.6], by choosing
$\varphi(s,t)=s^{p}t^{k+1-q}$, we obtain

######  Theorem A.2 ([IM23]).

Let $n\geq 2$ and $1\leq k\leq n-1$. Suppose $p\geq 1-k$ and $q\leq k+1$. Then
the smooth strictly convex origin-symmetric solution $M$ to (A.3) must be an
origin-centred sphere.

As an extension of the above theorem, by applying Lemma A.1 for $k<n$, we have
the following result:

######  Theorem A.3.

Let $n\geq 2$ and $1\leq k\leq n-1$. Suppose $p\geq 1-k$ and $q\leq
k+1+2\alpha_{*}$, where $\alpha_{*}$ is the maximum value satisfying the
following conditions:

(A.4) $\begin{split}\left\\{\begin{array}[]{l}0\leq\alpha\leq 1,\\\
-2\alpha(p+1)+(\alpha^{2}+2\alpha)(3p+1)\leq p+k-1,\\\
2\alpha(n+2)+(\alpha^{2}+2\alpha)(2n+k+6)\leq
p+k-1.\end{array}\right.\end{split}$

Then the smooth strictly convex origin-symmetric solution $M$ to (A.3) must be
an origin-centred sphere.

######  Remark A.1.

When $\alpha_{*}=0$, Theorem A.3 reduces to Theorem A.2.

###### Proof of Theorem A.3.

Suppose there exists a smooth strictly origin-symmetric convex solution $M$ to
(A.3). For convenience, we denote $\tau=q-k-1$. Then we have
$dV_{k}=u^{p}|X|^{-\tau}d\sigma$.

Since $\int_{\mathbb{S}^{n}}|X|^{\alpha}XdV_{k}=0$ for any
$\alpha\in\mathbb{R}$, it follows from (A.1) that for $\alpha\geq 0$

(A.5)
$\begin{split}k\int_{\mathbb{S}^{n}}|X|^{2\alpha+2}dV_{k}&\leq\int_{\mathbb{S}^{n}}|X|^{2\alpha}u\left(\Delta
u+nu-(k+1)\frac{\sigma_{k+1}}{\sigma_{k}}\right)dV_{k}\\\
&\quad+(\alpha^{2}+2\alpha)\int_{\mathbb{S}^{n}}|X|^{2\alpha-2}|\nabla
u|^{2}u\left(\Delta
u+nu-(k+1)\frac{\sigma_{k+1}}{\sigma_{k}}\right)dV_{k}.\end{split}$

Since $\alpha_{*}\geq 0$ and $\alpha_{*}-\frac{\tau}{2}\geq 0$, by using
(5.2), we obtain

(A.6) $\begin{split}&\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}}u\Delta
udV_{k}=\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-\tau}u^{p+1}\Delta ud\sigma\\\
=&-(p+1)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-\tau}u^{p}|\nabla
u|^{2}d\sigma-\left(\alpha_{*}-\frac{\tau}{2}\right)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-\tau}u^{p+1}\langle\nabla
u,\nabla|X|^{2}\rangle d\sigma\\\
\leq&-(p+1)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-\tau}u^{p}|\nabla
u|^{2}d\sigma\\\ =&-(p+1)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}}|\nabla
u|^{2}dV_{k}\end{split}$

and

(A.7) $\begin{split}&\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-2}|\nabla
u|^{2}u\Delta udV_{k}=\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-\tau-2}|\nabla
u|^{2}u^{p+1}\Delta ud\sigma\\\
=&-(p+1)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-\tau-2}u^{p}|\nabla
u|^{4}d\sigma-\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-\tau-2}u^{p+1}\langle\nabla
u,\nabla|\nabla u|^{2}\rangle d\sigma\\\
&\quad-\left(\alpha_{*}-\frac{\tau}{2}-1\right)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-\tau-4}u^{p+1}|\nabla
u|^{2}\langle\nabla u,\nabla|X|^{2}\rangle d\sigma\\\
=&-(p+1)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-\tau-2}u^{p}|\nabla
u|^{4}d\sigma+2\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-\tau-2}u^{p+2}|\nabla
u|^{2}d\sigma\\\
&\quad-\int_{\mathbb{S}^{n}}\left(|X|^{2}+\left(\alpha_{*}-\frac{\tau}{2}-1\right)|\nabla
u|^{2}\right)|X|^{2\alpha_{*}-\tau-4}u^{p+1}\langle\nabla
u,\nabla|X|^{2}\rangle d\sigma\\\
\leq&-(p+1)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-\tau-2}u^{p}|\nabla
u|^{4}d\sigma+2\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-\tau-2}u^{p+2}|\nabla
u|^{2}d\sigma\\\ =&-(p+1)\int_{\mathbb{S}^{n}}|X|^{2\alpha-2}|\nabla
u|^{4}dV_{k}+2\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-2}u^{2}|\nabla
u|^{2}dV_{k}.\end{split}$

Suppose that $\\{e_{i}\\}_{i=1}^{n}$ is a local orthonormal frame for
$\mathbb{S}^{n}$ such that
$(u_{ij}+u\delta_{ij})(z_{0})=\lambda_{i}(z_{0})\delta_{ij}$, then $\langle
X,X_{i}\rangle=\langle X,\lambda_{i}e_{i}\rangle=\lambda_{i}u_{i}$. It is
well-known that $\sigma_{k+1}^{ii}=\sigma_{k}-\lambda_{i}\sigma_{k}^{ii}$ at
$z_{0}$. Then we have at $z_{0}$

$\displaystyle
0\leq\sum_{i,j}u_{i}u_{j}\sigma_{k+1}^{ij}=\sum_{i}u_{i}^{2}(\sigma_{k}-\lambda_{i}\sigma_{k}^{ii})\leq|\nabla
u|^{2}\sigma_{k},$

and

$\displaystyle\sum_{i,j}\langle X,X_{i}\rangle
u_{j}\sigma_{k+1}^{ij}=\sum_{i}\lambda_{i}u_{i}^{2}(\sigma_{k}-\lambda_{i}\sigma_{k}^{ii})\leq|\nabla
u|^{2}\sigma_{1}\sigma_{k}.$

Note that $\nabla_{i}\sigma_{k+1}^{ij}=0$ and
$(k+1)\sigma_{k+1}=\sigma_{k+1}^{ij}(u_{ij}+u\delta_{ij})$. Since
$0\leq\alpha_{*}\leq 1$, by using (A.7), we calculate

(A.8)
$\begin{split}&(k+1)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}}u\frac{\sigma_{k+1}}{\sigma_{k}}dV_{k}\\\
=&(k+1)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}}u^{2}\sigma_{k+1}d\sigma=\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}}u^{2}\sigma_{k+1}^{ij}(u_{ij}+u\delta_{ij})d\sigma\\\
=&-2\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}}uu_{i}u_{j}\sigma_{k+1}^{ij}d\sigma-2\alpha_{*}\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-2}u^{2}\langle
X,X_{i}\rangle
u_{j}\sigma_{k+1}^{ij}d\sigma+\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}}u^{3}\sigma_{k+1}^{ij}\delta_{ij}d\sigma\\\
\geq&-2\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}}|\nabla
u|^{2}dV_{k}-2\alpha_{*}\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-2}u|\nabla
u|^{2}\sigma_{1}dV_{k}+(n-k)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}}u^{2}dV_{k}\\\
\geq&-2\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}}|\nabla
u|^{2}dV_{k}+2\alpha_{*}(p+1)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-2}|\nabla
u|^{4}dV_{k}\\\
&\quad-2\alpha_{*}(n+2)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-2}u^{2}|\nabla
u|^{2}dV_{k}+(n-k)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}}u^{2}dV_{k}\end{split}$

and

(A.9) $\begin{split}&(k+1)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-2}|\nabla
u|^{2}u\frac{\sigma_{k+1}}{\sigma_{k}}dV_{k}=\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-2}|\nabla
u|^{2}u^{2}\sigma_{k+1}^{ij}(u_{ij}+u\delta_{ij})d\sigma\\\
=&-2\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-2}|\nabla
u|^{2}uu_{i}u_{j}\sigma_{k+1}^{ij}d\sigma-2(\alpha_{*}-1)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-4}|\nabla
u|^{2}u^{2}\langle X,X_{i}\rangle u_{j}\sigma_{k+1}^{ij}d\sigma\\\
&\quad-\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-2}u^{2}(|\nabla
u|^{2})_{i}u_{j}\sigma_{k+1}^{ij}d\sigma+\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-2}|\nabla
u|^{2}u^{3}\sigma_{k+1}^{ij}\delta_{ij}d\sigma\\\
=&-2\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-2}|\nabla
u|^{2}uu_{i}u_{j}\sigma_{k+1}^{ij}d\sigma-2\int_{\mathbb{S}^{n}}\left((\alpha_{*}-1)|\nabla
u|^{2}+|X|^{2}\right)|X|^{2\alpha_{*}-4}u^{2}\langle X,X_{i}\rangle
u_{j}\sigma_{k+1}^{ij}d\sigma\\\
&\quad+2\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-2}u^{3}u_{i}u_{j}\sigma_{k+1}^{ij}d\sigma+(n-k)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-2}|\nabla
u|^{2}u^{2}dV_{k}\\\ \geq&-2\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-2}|\nabla
u|^{4}dV_{k}-2\int_{\mathbb{S}^{n}}\left((\alpha_{*}-1)|\nabla
u|^{2}+|X|^{2}\right)|X|^{2\alpha_{*}-4}u|\nabla u|^{2}\sigma_{1}dV_{k}\\\
&\quad+(n-k)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-2}|\nabla
u|^{2}u^{2}dV_{k}\\\ \geq&\ 2p\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-2}|\nabla
u|^{4}dV_{k}-(n+k+4)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-2}|\nabla
u|^{2}u^{2}dV_{k}.\end{split}$

Substituting (A.6), (A.7), (A.8) and (A.9) into (A.5), we obtain

(A.10)
$\begin{split}0&\leq\left(1-k-p-2\alpha_{*}(p+1)+(\alpha_{*}^{2}+2\alpha_{*})(3p+1)\right)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-2}|\nabla
u|^{4}dV_{k}\\\
&\quad+\left(1-k-p+2\alpha_{*}(n+2)+(\alpha_{*}^{2}+2\alpha_{*})(2n+k+6)\right)\int_{\mathbb{S}^{n}}|X|^{2\alpha_{*}-2}|\nabla
u|^{2}u^{2}dV_{k}.\end{split}$

According to the conditions (A.4), it can be deduced that the right hand side
of (A.10) is less than or equal to $0$. Hence $\nabla u\equiv 0$ on
$\mathbb{S}^{n}$, and then $M$ is an origin-centred sphere. We complete the
proof of Theorem A.3. ∎

## References