# Nonuniqueness of solutions to the $L_{p}$ chord Minkowski problem

Yuanyuan Li
School of Mathematical Sciences University of Science and Technology of China
Hefei 230026  China
E-mails<EMAIL_ADDRESS>
###### Abstract

This paper explores the nonuniqueness of solutions to the $L_{p}$ chord
Minkowski problem for negative $p.$ The $L_{p}$ chord Minkowski problem was
recently posed by Lutwak, Xi, Yang and Zhang, which seeks to determine the
necessary and sufficient conditions for a given finite Borel measure such that
it is the $L_{p}$ chord measure of a convex body, and it includes the chord
Minkowski problem and the $L_{p}$ Minkowski problem.

##  1 Introduction

The central objects in study of convex geometry are convex bodies. A convex
body in $n$-dimensional Euclidean space $\mathbb{R}^{n},$ is a compact convex
set with non-empty interior. The Brunn-Minkowski theory is a study of convex
bodies which centers around the study of geometric functionals and the
differential of these functionals. When geometric invariants arise as
geometric functionals of convex bodies, geometric measures are often viewed as
differentials of geometric invariants. One of the cornerstones of the Brunn-
Minkowski theory is the Minkowski problem. It is a problem of priscribing
geometric measure generated by convex bodies, which is concerned about
necessary and sufficient conditions for a given measure such that it arises as
the measure generated by a convex body. The most studied Minkowski-type
problem is the classical Minkowski problem, which focuses on the surface area
measures of convex bodies. For a comprehensive discussion on the Minkowski
problem and its resolution, we recommend readers consulting Pogorelov [20] and
Cheng–Yau [8].

Recently, a new family of geometric measures were introduced by Lutwak-XYZ[16]
by studying of a variational formula regarding intergral geometric invariants
of convex bodies called chord integrals. Let $K\in\mathcal{K}^{n}$ where
$\mathcal{K}^{n}:=\\{\text{all convex bodies in }\mathbb{R}^{n}\\},$ the $q$th
chord integral $I_{q}(K)$ is defined by

$I_{q}(K)=\int_{\mathcal{L}^{n}}|K\cap\ell|^{q}d\ell,$ (1.1)

where $\mathcal{L}^{n}$ denotes the Grassmannian of 1-dimensional affine
subspace of $\mathbb{R}^{n},$ $|K\cap\ell|$ denotes the length of the chord
$K\cap\ell,$ and the integration is with respect to Haar measure on the affine
Grassmannian $\mathcal{L}^{n},$ which is normalized to be a probability
measure when restricted to rotations and to be $(n-1)$-dimensional Lebesgue
measure when restricted to parallel translations.

$I_{1}(K)=V(k),\quad I_{0}(K)=\frac{\omega_{n-1}}{n\omega_{n}}S(K),\quad
I_{n+1}(K)=\frac{n+1}{\omega_{n}}V(K)^{2},$

where $\omega_{n}$ denotes the volume of $n$-dimensional unit ball. Note that
$I_{q}(B_{n})=\frac{2^{q}\omega_{n}\omega_{n+q-1}}{\omega_{q}},$ where $B_{n}$
is the n-dimensional unit ball. One can see from the above fomula that the
chord integrals include the convex body’s volume and surface area as two
special cases. These are Crofton’s volume formula, Cauchy’s integral formula
for surface area, and the Poincaré-Hadwiger formula, respectively (see [[21],
[25]]).

The chord measures and the Minkowski problems associated with chord measures
were posed in [16]. They showed that the chord measures are the differentials
of chord integrals and completely solved the chord Minkowski problem except
for the critical case of the Christoffel-Minkowski problem. The $q$th chord
measure is a finite Borel measure on $\mathbb{S}^{n-1}$ defined by

$F_{q}(K,\eta)=\frac{2q}{\omega_{n}}\int_{\nu^{-1}_{K}(\eta)}\tilde{V}_{q-1}(K,z)d\mathcal{H}^{n-1}(z),\text{
Borel }\eta\subset\mathbb{S}^{n-1},$ (1.2)

where $\widetilde{V}_{q-1}(K,z)$ is the $q-1$ th dual quermassintegral with
respect to $z$.(See (2.1).)

$F_{0}(K,\cdot)=\frac{(n-1)\omega_{n-1}}{n\omega_{n}}S_{n-2}(K,\cdot),\quad
F_{1}(K,\cdot)=S_{n-1}(K,\cdot),$

where $S_{i}(K,\cdot)$ is the $i$th order area measure of $K.$ Once chord
measures are constructed, the $L_{p}$ chord measures follow naturally by
extensions. For $K\in\mathcal{K}^{n}_{o}$ and $p\in\mathbb{R},$ the $L_{p}$
chord measures are defined by

$F_{p,q}(K,\eta)=\frac{2q}{\omega_{n}}\int_{\nu^{-1}_{K}(\eta)}(z\cdot\nu_{K}(z))^{1-p}\tilde{V}_{q-1}(K,z)d\mathcal{H}^{n-1}(z),\text{
Borel }\eta\subset\mathbb{S}^{n-1}.$ (1.3)

When $p=0,$ it is the cone-chord measure. When $q=1,$ $F_{p,1}(K,\cdot)$ is
the $L_{p}$ surface area measure. When $q=0,$ $F_{p,0}(K,\cdot)$ is the
$L_{p}-(n-2)$th area measure.

The $L_{p}$-Minkowski problem was first formulated and studied by Lutwak in
[17]. It has been rapidly attracting much attention; Lutwak introduced the
important $L_{p}$ surface area measure and its associated Minkowski problem in
the $L_{p}$ Brunn-Minkowski theory. Many cases of the $L_{p}$ Minkowski
problem have been solved. The logarithmic Minkowski problem is one of the most
central Minkowski type problems and is the problem of characterizing the cone-
volume measure; see B$\ddot{o}$r$\ddot{o}$czky, Lutwak, Yang and Zhang [[2],
[4]], Zhu [[30],[3]], Stancu [[22], [23]], Gage [12], Xi and Leng [28], Firey
[11], Andrews [1], Chen, Huang, Li and Liu [9], Chen, Feng, Liu [10], [[29],
[5]] and reference therein. The centro-affine Minkowski problem is unsolved,
see [7]. For more classical Brunn-Minkowski theory and its recent
developments, we suggest readers to Schneider’s book [24].

The $L_{p}$ chord Minkowski problem posed by Xi-LZY [16] is a problem of
prescribing the $L_{p}$ chord measure: Given a finite Borel measure $\mu$ on
$\mathbb{S}^{n-1},p\in\mathbb{R},$ and $q\geqslant 0.$ Asking what are the
necessary and sufficient conditions for $\mu$ such that $\mu$ is the $L_{p}$
chord measure of a convex body $K\in\mathcal{K}^{n}_{o},$ namely

$F_{p,q}(K,\cdot)=\mu$ (1.4)

when $p=1,$ it is the chord Minkowski problem. When $q=1,$ it is the $L_{p}$
Minkowski problem. When $\mu$ has a density $f$ that is an integrable
nonnegtive function on $\mathbb{S}^{n-1},$ equation(1.4) becomes a new type of
Monge-Ampère equation on $\mathbb{S}^{n-1}$:

$\mbox{det}(\nabla^{2}h+hI)=\frac{h^{p-1}f}{\widetilde{V}_{q-1}([h],\bar{\nabla}h)},\text{
on }\mathbb{S}^{n-1},$ (1.5)

where $\nabla^{2}h$ is the covariant differentiation of $h$ with respect to an
orthonormal frame on $\mathbb{S}^{n-1},$ we look for a solution $h$ which is
the support function for some nondegenerate convex body. We can extend $h$ to
$\mathbb{R}^{n}$ via homogeneity and $\bar{\nabla}h$ is the Euclidean gradient
of $h$ in $\mathbb{R}^{n},$ and $\widetilde{V}_{q-1}([h],\bar{\nabla}h)$ is
the $(q-1)$th dual quermassintegral of the Wulff-shape $[h]$ of $h$ with
repect to the point $\bar{\nabla}h.$

In their paper[16], Lutwak-XYZ gave a sufficient condition for the symmetric
case of the chord log-Minkowski problem by studying the delicate concentration
properties of cone-chord measures. Shortly thereafter, Xi, Yang, Zhang and
Zhao [26] solved the $L_{p}$ chord Minkowski problem for $p>1$ and for $0<p<1$
under the symmetric condition, where the origin symmetry played a crucial role
in the case of $0\leqslant p<1.$ More recently, Xi, Guo and Zhao solved the
$L_{p}$ chord Minkowski problem when $0\leqslant p<1$, without any symmetry
assumptions. We solve the $L_{p}$ chord Minkowski problem when $p<0$, without
any symmetry assumptions. Lately, we solved the $L_{p}$ chord Minkowski
problem in the case of discrete measures whose supports are in general
position for negative $p$ and $q>0.$ As for general Borel measure with a
density, we also give a proof but need $p\in(-n,0)$ and $n+1>q\geqslant 1,$
without any symmetry assumptions.

The aim of this paper is to establish some nonuniqueness for the $L_{p}$ chord
Minkowski problem for $p<0<q.$ As far as we know, this is the first result
towards uniqueness and nonuniqueness of solution to the $L_{p}$ chord
Minkowski problem. Our main theorem is as following:

######  Theorem 1.1.

For $p<0,2<q<n+1,$ there exists a positive function $f\in
C^{\infty}(\mathbb{S}^{n-1})$ such that (1.5) admits at least two different
solutions.

######  Remark 1.2.

We have to note that the method in this paper is inspired by X.-J. Wang et
al[14] and Q.-R. Li et al[18]. But the situation in our case is more
complicated, since the dual quermassintegral $\widetilde{V}_{q-1}(K,z)$ is a
nonlocal term in the integrand of $I_{q}(K).$ And we also note that the $q-1$
th dual quermassintegral $\widetilde{V}_{q}(K,z)$ of $K$ with respect to
$z\in\partial K$ is more delicate than the $q-1$ th dual quermassintegral
$\widetilde{V}_{q}(K)$ of $K\in\mathcal{K}^{n}_{o}.$

To prove the Theorem 1.1, we need to find at least two different solutions.
One is constructed from the solution of classic Minkowski problem with a
special right-hand-side, such that the solution to (1.5) obtained in this way
has its $q-$th chord integral as small as we want. The other solution is from
a new existence result for equation (1.5) by the variational method. Before we
state the result, we first need to introduce some notations.

######  Definition 1.3.

A function $f:\mathbb{S}^{n-1}\rightarrow\mathbb{R}$ is called rotationally
symmetric if it satisfies

$f(Ax^{\prime},x_{n})=f(x^{\prime},x_{n}),\quad\forall
x=(x^{\prime},x_{n})\in\mathbb{S}^{n-1}\text{ and }A\in O(n-1),$

where $x^{\prime}=(x_{1},\cdots,x_{n-1})$ and $O(\cdot)$ denotes the
orthogonal group.

Then, here comes our new existence result of solutions to (1.5).

######  Theorem 1.4.

For $p<0,1\leqslant q<n+1,$ and $\alpha,\beta$ satisfying

$\begin{gathered}\alpha>\max\\{1-n,1-n+\frac{2-n-q}{n+q-1}p\\},\\\
\beta>\max\\{-1,-1-\frac{p}{n+q-1}\\}.\end{gathered}$

If $f$ is a nonnegative, rotationally symmetric, even function on
$\mathbb{S}^{n-1}$ and satisfies

$f\leqslant
C\left|x^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta},\|f\|_{L^{1}(\mathbb{S}^{n-1})}>0$
(1.6)

for some positive constant $C.$ Then there exists a rotationally symmetric
even solution to (1.5). Moreover, we have its chord integral as follows:

$I_{q}(K)\geqslant c>0,$ (1.7)

where $c$ depends only on $n,p,q,\alpha,\beta.$

######  Remark 1.5.

In our previous paper, we solved the $L_{p}$ chord Minkowski problem for
general Borel measure with a density $f$, without any symmetry assumptions ,
but need $p\in(-n,0)$ and $n+1>q\geqslant 1.$ Here, when $f$ satisfies the
conditions in Theorem 1.4, we can obtain the existence result for all $p<0.$
While the proof of the two existence results follows similar approaches, we
would like to emphasize that the proof of $C^{0}$ estimate is totally
different and more difficult.

The remainder of this paper is structured as follows. In Section 2, we present
fundamental concepts in the theory of convex bodies and integral geometry. In
Section 3, we will construct a smooth, positive function $f,$ which is
rotationally symmetric in $x_{1},\cdots,x_{n-1}$ and even, such that (1.5) has
a solution with small chord integral. In Section 4, we establish a variational
solution and prove Theorem 1.4. In Section 5, we prove Theorem 1.1 based on
the existence result in Theorem 1.4 and the solution constructed in Section 3.

##  2 Preliminaries

In this section, our objective is to establish notations and gather relevant
results from the literature that will be necessary for the subsequent
analysis.

We denote $x\cdot y$ as the standard inner product of $x,y\in\mathbb{R}^{n},$
and write $|x|=\sqrt{x\cdot x}$ for the Euclidean norm of $x.$ We write
$\mathbb{S}^{n-1}$ as $(n-1)$-dimension unit sphere of $\mathbb{R}^{n},$ and
denote $\mathcal{H}^{n-1}$ as the $(n-1)$-dimensional spherical Lebesgue
measure. Denote $\mathcal{K}^{n}$ for the collection of all convex bodies in
$\mathbb{R}^{n}$ and $\mathcal{K}^{n}_{o}$ for the subset of $\mathcal{K}^{n}$
that contains the origin in the interior.

Let $\Omega\subset\mathbb{S}^{n-1}$ be a closed set of the unit sphere, not
lying in a closed hemisphere, and a positive continuous function
$h:\mathbb{S}^{n-1}\rightarrow\mathbb{R}$ is given.(Only the values of h on
$\Omega$ will be needed, but without loss of generality we may assume that $h$
is defined on all of $\mathbb{S}^{n-1}.$) The Wulff shape of $h$ is defined by

$[h]=\\{x\in\mathbb{R}^{n}:x\cdot u\leqslant h(u)\text{ for all
}u\in\mathbb{S}^{n-1}\\}.$

Let $K\in\mathcal{K}^{n},$ $h(v)=h_{K}(v)=\max\\{v\cdot x,x\in K\\},$
$\rho(u)=\rho_{K}(u)=\max\\{\lambda:\lambda u\in K\\}$ are the support
function and the radial function of convex body $K$ defined from
$\mathbb{S}^{n-1}\rightarrow\mathbb{R}.$ We write the support hyperplane of
$K$ with the outer unit normal $v$ as

$H_{K}(v)=\left\\{x\in\mathbb{R}^{n}:x\cdot v=h(v)\right\\},$

the half-space $H^{-}(K,v)$ in direction $v$ is defined by

$H^{-}_{K}(v)=\left\\{x\in\mathbb{R}^{n}:x\cdot v\leqslant h(v)\right\\}.$

Denote $\partial K$ as the boundary of $K$, that is, $\partial
K=\\{\rho_{K}(u)u:u\in\mathbb{S}^{n-1}\\}.$ The spherical image
$\nu=\nu_{K}:\partial K\rightarrow\mathbb{S}^{n-1}$ is given by

$\nu(x)=\\{v\in\mathbb{S}^{n-1}:x\in H_{K}(v)\\},$

let $\sigma_{K}\subset\partial K$ denote the set of all points $x\in\partial
K,$ such that the set $\nu_{K}(x)$ contains more than one element.
Fortunately, we have $\mathcal{H}^{n-1}(\sigma_{K})=0$ (see [24, page 84] )
and the radial Gauss image $\alpha=\alpha_{K}$ and the reverse radial Gauss
image $\alpha^{*}=\alpha^{*}_{K}$ are respectively defined by

$\alpha(\omega)=\\{\nu(\rho_{K}(u)u):u\in\omega\\},\alpha^{*}(\omega)=\\{u\in\mathbb{S}^{n-1}\nu(\rho_{K}(u)u)\in\omega\\}.$

Let $K\in\mathcal{K}^{n}$, for $z\in\operatorname{int}K$ and $q\in\mathbb{R}$,
the $q$ th dual quermassintegral $\widetilde{V}_{q}(K,z)$ of $K$ with respect
to $z$ is defined by

$\widetilde{V}_{q}(K,z)=\frac{1}{n}\int_{S^{n-1}}\rho_{K,z}(u)^{q}\mathrm{~{}d}u$
(2.1)

where $\rho_{K,z}(u)=\max\\{\lambda>0:z+\lambda u\in K\\}$ is the radial
function of $K$ with respect to $z$. When $z\in\partial
K,\widetilde{V}_{q}(K,z)$ is defined in the way that the integral is only over
those $u\in S^{n-1}$ such that $\rho_{K,z}(u)>0$. In another word,

$\widetilde{V}_{q}(K,z)=\frac{1}{n}\int_{\rho_{K,z}(u)>0}\rho_{K,z}(u)^{q}\mathrm{~{}d}u\text{,
whenever }z\in\partial K.$

In this case, for $\mathcal{H}^{n-1}$-almost all $z\in\partial K$, we have

$\widetilde{V}_{q}(K,z)=\frac{1}{2n}\int_{S^{n-1}}X_{K}(z,u)^{q}\mathrm{~{}d}u$

where the parallel $X$-ray of $K$ is the nonnegative function on
$\mathbb{R}^{n}\times S^{n-1}$ defined by

$X_{K}(z,u)=|K\cap(z+\mathbb{R}u)|,\quad z\in\mathbb{R}^{n},\quad u\in
S^{n-1}.$

When $q>0$, the dual quermassintegral is the Riesz potential of the
characteristic function, that is,

$\widetilde{V}_{q}(K,z)=\frac{q}{n}\int_{K}|x-z|^{q-n}\mathrm{~{}d}x$

Note that this immediately allows for an extension of
$\widetilde{V}_{q}(K,\cdot)$ to $\mathbb{R}^{n}$. An equivalent definition via
radial function can be found in [16]. By a change of variables, we obtain:

$\widetilde{V}_{q}(K,z)=\frac{q}{n}\int_{K-z}|y|^{q-n}\mathrm{~{}d}y$

since when $q>0$, the integrand $|y|^{q-n}$ being locally integrable, it can
be inferred that the dual quermassintegral $\widetilde{V}_{q}(K,z)$ is
continuous in $z$. Let $K\in\mathcal{K}^{n}$. The $X$-ray $X_{K}(x,u)$ and the
radial function $\rho_{K,z}(u)$ are related as follows:

$X_{K}(x,u)=\rho_{K,z}(u)+\rho_{K,z}(-u),\quad\text{ when }\quad
K\cap(x+\mathbb{R}u)=K\cap(z+\mathbb{R}u)\neq\varnothing.$

When $z\in\partial K$, then either $\rho_{K,z}(u)=0$ or $\rho_{K,z}(-u)=0$ for
almost all $u\in S^{n-1}$, and thus

$X_{K}(z,u)=\rho_{K,z}(u),\quad\text{ or }X_{K}(z,u)=\rho_{K,z}(-u),\quad
z\in\partial K\text{, }$

for almost all $u\in S^{n-1}$. Then, the chord integral $I_{q}(K)$ can be
represented as follows:

$I_{q}(K)=\frac{1}{n\omega_{n}}\int_{S^{n-1}}\int_{u^{\bot}}X_{K}(x,u)^{q}\mathrm{~{}d}x\mathrm{~{}d}u,\quad
q\geq 0.$

An elementary property of the functional $I_{q}$ is its homogeneity. If
$K\in\mathcal{K}^{n}$ and $q\geq 0$, then

$I_{q}(tK)=t^{n+q-1}I_{q}(K),$

for $t>0$. By compactness of $K$, it is simple to see that the chord integral
$I_{q}(K)$ is finite whenever $q\geq 0$. Let $K\in\mathcal{K}^{n}$ and $q>0$,
the chord measure $F_{q}(K,\cdot)$ is a finite Borel measure on $S^{n-1}$,
which can be expressed as:

$F_{q}(K,\eta)=\frac{2q}{\omega_{n}}\int_{v^{-1}(\eta)}\widetilde{V}_{q-1}(K,z)\mathrm{d}\mathcal{H}^{n-1}(z),\quad\text{
for each Borel }\eta\subset S^{n-1}.$

The mapping $v_{K}$ of $K$ is almost everywhere defined on $\partial K$ with
respect to the $(n-1)$-dimensional Hausdorff measure, owing to the convexity
of $K$. The chord measure $F_{q}(K,\cdot)$ is significant as it is obtained by
differentiating the chord integral $I_{q}$ in a certain sense, as shown in
(2.2). It is evident that the chord measure $F_{q}(K,\cdot)$ is absolutely
continuous with respect to the surface area measure $S_{n-1}(K,\cdot)$. In
[16, Theorem 4.3], it was demonstrated that:

$I_{q}(K)=\frac{1}{n+q-1}\int_{s^{n-1}}h_{K}(v)\mathrm{d}F_{q}(K,v)$

When $q>0$, a useful integral formula demonstrated in [16, Lemma 5.3] is

$2n\int_{\partial
K}\widetilde{V}_{q-1}(K,z)g\left(v_{K}(z)\right)\mathrm{d}\mathcal{H}^{n-1}(z)=\int_{s^{n-1}}\int_{\partial
K}X_{K}(z,u)^{q-1}g\left(v_{K}(z)\right)\mathrm{d}\mathcal{H}^{n-1}(z)\mathrm{d}u,$

for any $g\in C\left(S^{n-1}\right)$. Therefore, for each
$K\in\mathcal{K}^{n}$, we have

$\displaystyle\int_{S^{n-1}}g(v)\mathrm{d}F_{q}(K,v)$
$\displaystyle=\frac{q}{n\omega_{n}}\int_{S^{n-1}}\int_{\partial
K}X_{K}(z,u)^{q-1}g\left(v_{K}(z)\right)\mathrm{d}\mathcal{H}^{n-1}(z)\mathrm{d}u$
$\displaystyle=\frac{q}{n\omega_{n}}\int_{S^{n-1}}\int_{S^{n-1}}X_{K}\left(\rho_{K}(w)w,u\right)^{q-1}h_{K}\left(\alpha_{K}(w)\right)^{-1}$
$\displaystyle\quad\rho_{K}(w)^{n}g\left(\alpha_{K}(w)\right)\mathrm{d}w\mathrm{~{}d}u.$

Here, we denote $\rho_{K}=\rho_{K,o}$. For each $p\in\mathbb{R}$ and
$K\in\mathcal{K}_{o}^{n}$, the $L_{p}$ chord measure $F_{p,q}(K,\cdot)$ is
defined as follows:

$\mathrm{d}F_{p,q}(K,v)=h_{K}(v)^{1-p}\mathrm{~{}d}F_{q}(K,v)$

and we have an important property of $F_{p,q},$ its homogeneity, namely

$F_{p,q}(tK,\cdot)=t^{n+q-p-1}F_{p,q}(K,\cdot)$

for each $t>0.$

From Theorem 2.2 in [26], we know that if
$K_{i}\in\mathcal{K}^{n}_{o}\rightarrow K_{0}\in\mathcal{K}^{n}_{o},$ then the
chord measure $F_{q}(K_{i},\cdot)$ converges to $F_{q}(K,\cdot)$ weakly.
Hence, one can immediately obtain that

$F_{p,q}(K_{i},\cdot)\rightarrow F_{p,q}(K,\cdot)\text{ weakly. }$

It was shown in [16] that the differential of the chord integral $I_{q}$ with
respect to the $L_{p}$ Minkowski combinations leads to the $L_{p}$ chord
measure: for $p\neq 0$,

$\left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}I_{q}\left(K+_{p}t\cdot
L\right)=\frac{1}{p}\int_{S^{n-1}}h_{L}^{p}(v)\mathrm{d}F_{p,q}(K,v),$

where $K+_{p}t\cdot L$ is the $L_{p}$ Minkowski combination between $K$ and
$L.$

Since we will use a variational method to solve the $L_{p}$ chord Minkowski
problem, the variational formula for chord integral is crucial and it is the
key to tansforming the Minkowski problem into the Lagrange equation of an
optimization problem.

######  Theorem 2.1 (Theorem 5.5 in [16]).

Let $q>0,$ and $\Omega$ be a compact subset of $\mathbb{S}^{n-1}$ that is not
contained in any closed hemisphere. Suppose that
$g:\Omega\rightarrow(0,\infty)$ is a family of continuous functions given by

$h_{t}=h_{0}+tg+o(t,\cdot),$

for each $t\in(-\delta,\delta)$ for some $\delta>0.$ Here $o(t,\cdot)\in
C(\Omega)$ and $o(t,\cdot)/v$ tends to $0$ uniformly on $\Omega$ as
$t\rightarrow 0.$ Let $K_{t}$ be the Wulff shape generated by $h_{t}$ and $K$
be the Wulff shape generated by $h_{0}.$ Then,

$\frac{d}{dt}\big{|}_{t=0}I_{q}(K_{t})=\int_{\Omega}g(v)dF_{q}(K,v).$ (2.2)

See also in [26, Theorem 2.1].

To solve the maximization problem posed in Section 4, delicate estimates for
chord integrals are needed. We collect the following lemma obtained in [16].

######  Lemma 2.2 (lemma 7.3 [16]).

Suppose $q\in(1,n+1)$ is not an integer. If $E$ is the ellipsoid in
$\mathbb{R}^{n}$ given by

$E=E\left(a_{1},\ldots,a_{n}\right)=\left\\{x\in\mathbb{R}^{n}:\frac{\left(x\cdot
e_{1}\right)^{2}}{a_{1}^{2}}+\cdots+\frac{\left(x\cdot
e_{n}\right)^{2}}{a_{n}^{2}}\leq 1\right\\}$

with $0<a_{1}\leq a_{2}\leq\cdots\leq a_{n}\leq 1$, then for any real $q$ and
integer $m$ such that $1\leq m<q<$ $m+1\leq n+1$,

$I_{q}(E)\leq c_{q,m,n}\left(a_{1}\cdots
a_{m}\right)^{2}a_{m}^{q-m-1}a_{m+1}\cdots a_{n}$

where $c_{q,m,n}$ is a constant that depends only on $q$ and $n$ (since
$m=\lfloor q\rfloor$ ) and is given by

$c_{q,m,n}=\begin{cases}\frac{2^{q-n+2}q(q-1)\omega_{n-1}^{2}}{(q-n)(q-n+1)n\omega_{n}}&m=n\\\
\frac{2^{n-m+3}q(q-1)(n-m)\omega_{m-1}^{2}\omega_{n-m}^{2}}{(m+1-q)(q-m)(q-m+1)n\omega_{n}}&m<n.\end{cases}$

As for $q\in\\{1,\cdots,n\\},$ we shall deduce the same form of estimate.
First, we recall a significant inequality presented in the following lemma,
which was obtained in [16]:

######  Lemma 2.3 (claim 8.1 [16]).

If $K\in\mathcal{K}^{n}_{o}$ and $1\leqslant r<s,$ then

$I_{r}(K)\leqslant c(s,r)V(K)^{1-\frac{r-1}{s-1}}I_{s}(K)^{\frac{r-1}{s-1}},$

with $c(s,r)=rs^{-\frac{r-1}{s-1}}.$

This inequality follows from a simple argument using jensen’s inequality. And
one immediately obtains the desired estimate for chord integrals when $q$ is
an integer.

##  3 Structural solution

Let $0<\epsilon<\frac{1}{2}$, $M_{\epsilon}\in GL(n)$ be given by

$M_{\epsilon}=diag(\epsilon,\cdots,\epsilon,1)=\left(\begin{array}[]{cc}\epsilon
I&0\\\ 0&1\end{array}\right),$

where $I$ is the unit $(n-1)\times(n-1)$ matrix.

Consider the following equation:

$\operatorname{det}\left(\nabla^{2}h+hI\right)(x)=\left|x^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}\left|M_{\epsilon}x\right|^{\gamma-\beta},\quad
x\in\mathbb{S}^{n-1}.$ (3.1)

We can choose appropriate indices $\alpha$, $\beta$ and $\gamma,$ in this
section, we need $2<q\leqslant 1+n,-1<\gamma<-1-\frac{p}{n+q-1},$ $\alpha$ and
$\beta$ are nonnegative and satisfying the assumptions of Theorem 1.4. As its
right-hand side is even with respect to the origin and satisfies the necessary
condition

$\int_{S^{n-1}}x_{k}\left|x^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}\left|M_{\epsilon}x\right|^{\gamma-\beta}=0$

for all $1\leqslant k\leqslant n$, this classical Minkowski problem exists a
solution $h_{\epsilon}$, which is unique up to translation by [8].

Let $h_{\epsilon}$ be the unique solution such that its associated convex body
$K_{h_{\epsilon}}$ centred at the origin. We have to note that
$K_{h_{\epsilon}}$ is rotationally symmetric and even and the positive
constants $C,\tilde{C}$ and $c_{i},C_{i}$ in the following context depend only
on $n,p,q,\alpha,\beta$, and $\gamma$ but independent of $\epsilon$.

######  Lemma 3.1.

Let $\gamma>-1.$ There exists a positive constant $C$, independent of
$\epsilon\in(0,\frac{1}{2})$, such that

$C^{-1}\leq h_{\epsilon}\leq C\text{ on }\quad\mathbb{S}^{n-1}.$ (3.2)

###### Proof.

By (3.1), $-1<\gamma,$ one has

$\displaystyle\operatorname{area}\left(\partial K_{h_{\epsilon}}\right)$
$\displaystyle=\int_{\mathbb{S}^{n-1}}\operatorname{det}\left(\nabla^{2}h+hI\right)(x)dx$
$\displaystyle=\int_{\mathbb{S}^{n-1}}\left|x^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}\left(\epsilon^{2}\left|x^{\prime}\right|^{2}+\left|x_{n}\right|^{2}\right)^{\frac{\gamma-\beta}{2}}dx$
$\displaystyle\leqslant\int_{\mathbb{S}^{n-1}}\left(\epsilon^{2}\left|x^{\prime}\right|^{2}+\left|x_{n}\right|^{2}\right)^{\gamma/2}dx$
$\displaystyle\leqslant\left\\{\begin{array}[]{lll}\int_{\mathbb{S}^{n-1}}\left|x_{n}\right|^{\gamma/2}\mathrm{~{}d}x&\text{
when }-1<\gamma<0\\\ \int_{\mathbb{S}^{n-1}}\mathrm{~{}d}x&\text{ when
}0\leqslant\gamma\end{array}\right.$ $\displaystyle\leq C,$

where $C$ is a positive constant depending on $n,\gamma$ but independent of
$\epsilon$. By John’s lemma,

$\frac{1}{n}E_{\epsilon}\subset K_{h_{\epsilon}}\subset E_{\epsilon}$

where $E_{\epsilon}$ is the minimum ellipsoid of $K_{h_{\epsilon}}$. Then,
this implies that

$\frac{1}{n}h_{E_{\epsilon}}\leq h_{\epsilon}\leq h_{E_{\epsilon}}\quad\text{
on }\quad\mathbb{S}^{n-1},$

where $h_{E_{\epsilon}}$ is the support function of $E_{\epsilon}$. Since
$K_{h_{\epsilon}}$ is rotationally symmetric and even, we have that
$E_{\epsilon}$ is also rotationally symmetric and even and centred at the
origin. Let $r_{1\epsilon},\cdots,r_{n\epsilon}$ be the lengths of the semi-
axes of $E_{\epsilon}$ along the $x_{1},\cdots,x_{n}$ axes. Then,
$r_{1\epsilon}=\cdots=r_{n-1;\epsilon}$ and

$h_{E_{\epsilon}}(x)=\sqrt{r_{1\epsilon}^{2}\left|x^{\prime}\right|^{2}+r_{n\epsilon}^{2}x_{n}^{2}},\quad\forall
x\in\mathbb{S}^{n-1}.$

By (3.1), one has

$\displaystyle{\mbox{Vol}}(K_{h_{\epsilon}})$
$\displaystyle=\frac{1}{n}\int_{\mathbb{S}^{n-1}}h_{\epsilon}\operatorname{det}\left(\nabla^{2}h_{\epsilon}+h_{\epsilon}I\right)(x)dx$
$\displaystyle=\frac{1}{n}\int_{\mathbb{S}^{n-1}}h_{\epsilon}\left|x^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}\left(\epsilon^{2}\left|x^{\prime}\right|^{2}+\left|x_{n}\right|^{2}\right)^{\frac{\gamma-\beta}{2}}dx$
$\displaystyle\geqslant\frac{1}{n}\int_{\mathbb{S}^{n-1}}h_{\epsilon}\left|x^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}\left(\epsilon^{2}\left|x^{\prime}\right|^{2}+\left|x_{n}\right|^{2}\right)^{\frac{\gamma}{2}}dx$
$\displaystyle\geqslant\frac{1}{n}\left\\{\begin{array}[]{lll}\int_{\mathbb{S}^{n-1}}h_{\epsilon}(x)\left|x^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}\mathrm{d}x&\text{
when }&-1<\gamma<0,\\\
\int_{\mathbb{S}^{n-1}}h_{\epsilon}(x)\left|x^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta+\gamma}\mathrm{~{}d}x&\text{
when }&\gamma\geqslant 0.\end{array}\right.$

Since

$h_{\epsilon}(x)\geq\frac{1}{\sqrt{2}n}\left(r_{1\epsilon}\left|x^{\prime}\right|+r_{n\epsilon}\left|x_{n}\right|\right),\quad\forall
x\in\mathbb{S}^{n-1}.$

Recall that $\alpha,\beta$ are nonnegative, then we have

$\displaystyle{\mbox{Vol}}(K_{h_{\epsilon}})$
$\displaystyle\geqslant\frac{1}{n}\left\\{\begin{array}[]{lll}\int_{\mathbb{S}^{n-1}}\frac{1}{\sqrt{2}n}\left(r_{1\epsilon}\left|x^{\prime}\right|+r_{n\epsilon}\left|x_{n}\right|\right)\left|x^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}\mathrm{d}x&\text{
when }&-1<\gamma<0,\\\
\int_{\mathbb{S}^{n-1}}\frac{1}{\sqrt{2}n}\left(r_{1\epsilon}\left|x^{\prime}\right|+r_{n\epsilon}\left|x_{n}\right|\right)\left|x^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta+\gamma}\mathrm{~{}d}x&\text{
when }&\gamma\geqslant 0.\end{array}\right.$ $\displaystyle\geqslant
c(r_{1\epsilon}+r_{n\epsilon}),$

therefore, by the isoperimetric inequality, we obtain that

$\operatorname{vol}\left(K_{h_{\epsilon}}\right)\leq
C_{n}\operatorname{area}\left(\partial
K_{h_{\epsilon}}\right)^{\frac{n}{n-1}}\leq C,$

and hence

$\max h_{\epsilon}<r_{1\epsilon}+r_{n\epsilon}\leqslant
c{\mbox{Vol}}(K_{h_{\epsilon}})\leqslant C.$ (3.3)

On the other hand, we have

$\displaystyle\operatorname{vol}\left(K_{h_{\epsilon}}\right)$
$\displaystyle\leq\operatorname{vol}\left(E_{\epsilon}\right)$
$\displaystyle=\kappa_{n}r_{1\epsilon}^{n-1}r_{n\epsilon}$ $\displaystyle\leq
C_{n}\left(\max h_{\epsilon}\right)^{n-1}\cdot\min h_{\epsilon}$

where $\kappa_{n}$ is the volume of the unit ball in $\mathbb{R}^{n}$. Since

$\max h_{\epsilon}\leqslant c{\mbox{Vol}}(K_{h_{\epsilon}})\leq C\left(\max
h_{\epsilon}\right)^{n-1}\cdot\min h_{\epsilon}$

namely

$1\leq C\left(\max h_{\epsilon}\right)^{n-2}\cdot\min h_{\epsilon}$

By (3.3), we obtain the bound of $h_{\epsilon}$ from below. Now, the proof of
this lemma is completed. ∎

Define

$H_{\epsilon}(x):=\epsilon^{\frac{n-p-4-\gamma+q}{n-p+q-1}}\left|M_{\epsilon}^{-1}x\right|\cdot
h_{\epsilon}\left(\frac{M_{\epsilon}^{-1}x}{\left|M_{\epsilon}^{-1}x\right|}\right),\quad
x\in\mathbb{S}^{n-1}.$ (3.4)

######  Lemma 3.2.

The function $H_{\epsilon}$ satisfies the equation

$\mbox{det}(\nabla^{2}H_{\epsilon}+H_{\epsilon})=\frac{H_{\epsilon}^{p-1}f_{\epsilon}}{\widetilde{V}_{q-1}([H_{\epsilon}],\bar{\nabla}H_{\epsilon})},\text{
on }\mathbb{S}^{n-1},$ (3.5)

where

$f_{\epsilon}(x):=h_{\epsilon}\left(x_{\epsilon}\right)^{1-p}\left|X^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}\left|N_{\epsilon}X\right|^{-\gamma-\alpha-
n-p}\frac{1}{n}\int_{S^{n-1}}\left|N_{\epsilon}y\right|^{q-1-n}\rho_{K_{h_{\epsilon}},\bar{\nabla}h_{\epsilon}(x_{\epsilon})}^{q-1}(y)dy,$
(3.6)

and

$x_{\epsilon}=\frac{M_{\epsilon}^{-1}x}{\left|M_{\epsilon}^{-1}x\right|}\quad\text{
and }\quad N_{\epsilon}=\epsilon
M_{\epsilon}^{-1}=\left(\begin{array}[]{cc}I&0\\\
0&\epsilon\end{array}\right).$

###### Proof.

Let

$u_{\epsilon}(x):=\left|M_{\epsilon}^{-1}x\right|\cdot
h_{\epsilon}\left(\frac{M_{\epsilon}^{-1}x}{\left|M_{\epsilon}^{-1}x\right|}\right)$

By the invariance of the quantity
$h_{\epsilon}^{n+1}\operatorname{det}\left(\nabla^{2}h_{\epsilon}+h_{\epsilon}I\right)$
under linear transformations, see [[7], Proposition 7.1] or formula (2.12) in
[19], we have

$\operatorname{det}\left(\nabla^{2}u_{\epsilon}+u_{\epsilon}I\right)(x)=\operatorname{det}\left(\nabla^{2}h_{\epsilon}+h_{\epsilon}I\right)\left(\frac{M_{\epsilon}^{-1}x}{\left|M_{\epsilon}^{-1}x\right|}\right)\cdot\frac{\left(\operatorname{det}M_{\epsilon}^{-1}\right)^{2}}{\left|M_{\epsilon}^{-1}x\right|^{n+1}}.$
(3.7)

Since

$x_{\epsilon}=\frac{M_{\epsilon}^{-1}x}{\left|M_{\epsilon}^{-1}x\right|}=\frac{\left(\epsilon^{-1}x^{\prime},x_{n}\right)}{\left|M_{\epsilon}^{-1}x\right|}=\frac{\left(x^{\prime},\epsilon
x_{n}\right)}{\left|N_{\epsilon}x\right|}.$

By (3.1), we then have

$\displaystyle\operatorname{det}\left(\nabla^{2}h_{\epsilon}+h_{\epsilon}I\right)\left(\frac{M_{\epsilon}^{-1}x}{\left|M_{\epsilon}^{-1}x\right|}\right)$
$\displaystyle=\frac{\left|x^{\prime}\right|^{\alpha}}{\left|N_{\epsilon}x\right|^{\alpha}}\cdot\frac{\left|\epsilon
x_{n}\right|^{\beta}}{\left|N_{\epsilon}x\right|^{\beta}}\cdot\left(\frac{1}{\left|M_{\epsilon}^{-1}x\right|}\right)^{\gamma-\beta}$
$\displaystyle=\frac{\left|x^{\prime}\right|^{\alpha}}{\left|N_{\epsilon}x\right|^{\alpha}}\cdot\frac{\left|\epsilon
x_{n}\right|^{\beta}}{\left|N_{\epsilon}x\right|^{\beta}}\cdot\left(\frac{\epsilon}{\left|N_{\epsilon}x\right|}\right)^{\gamma-\beta}$
$\displaystyle=\epsilon^{\gamma}\left|x^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}\left|N_{\epsilon}x\right|^{-\gamma-\alpha}.$

Applying this into (3.7), we obtain

$\displaystyle\operatorname{det}\left(\nabla^{2}u_{\epsilon}+u_{\epsilon}I\right)(x)$
$\displaystyle=\epsilon^{\gamma}\left|x^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}\left|N_{\epsilon}x\right|^{-\gamma-\alpha}\cdot\frac{\left(\epsilon^{1-n}\right)^{2}\epsilon^{n+1}}{\left|N_{\epsilon}x\right|^{n+1}}$
$\displaystyle=\epsilon^{\gamma+3-n}\left|x^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}\left|N_{\epsilon}x\right|^{-\gamma-\alpha-n-1}.$

By the definition of $u_{\epsilon}$, we have

$\begin{gathered}u_{\epsilon}(x)=\epsilon^{-1}\left|N_{\epsilon}x\right|\cdot
h_{\epsilon}\left(x_{\epsilon}\right),\\\
\left(\bar{\nabla}u_{\epsilon}\right)(x)=M_{\epsilon}^{-T}\left(\bar{\nabla}h_{\epsilon}\right)\left(x_{\epsilon}\right)=\epsilon^{-1}N_{\epsilon}\left(\bar{\nabla}h_{\epsilon}\right)\left(x_{\epsilon}\right)\end{gathered}$

then

$\displaystyle\rho_{K_{u_{\epsilon}},\bar{\nabla}u_{\epsilon}(x)}(y)$
$\displaystyle=\max\\{\lambda\in\mathbb{R}:\lambda y\in
K_{u_{\epsilon}}-\bar{\nabla}u_{\epsilon}(x)\\}$
$\displaystyle=\max\\{\lambda\in\mathbb{R}:\lambda y\in
M_{\epsilon}^{-1}K_{h_{\epsilon}}-M_{\epsilon}^{-1}M_{\epsilon}\bar{\nabla}u_{\epsilon}(x)\\}$
$\displaystyle=\max\\{\lambda\in\mathbb{R}:M_{\epsilon}^{-1}M_{\epsilon}\lambda
y\in
M_{\epsilon}^{-1}(K_{h_{\epsilon}}-M_{\epsilon}\bar{\nabla}u_{\epsilon}(x))\\}$
$\displaystyle=\rho_{K_{h_{\epsilon}},M_{\epsilon}(\bar{\nabla}u_{\epsilon}(x))}(M_{\epsilon}(y))$
$\displaystyle=\rho_{K_{h_{\epsilon}},\bar{\nabla}h_{\epsilon}(x_{\epsilon}))}(M_{\epsilon}(y))$

Hence

$\displaystyle\widetilde{V}_{q-1}\left(K_{u_{\epsilon}},\bar{\nabla}u_{\epsilon}(x)\right)$
$\displaystyle=\frac{1}{n}\int_{\mathbb{S}^{n-1}}\left(\rho_{K_{u_{\epsilon}},\bar{\nabla}u_{\epsilon}(x)}(u)\right)^{q-1}du$
(3.8)
$\displaystyle=\frac{1}{n}\int_{\mathbb{S}^{n-1}}\left(\left|M_{\epsilon}u\right|^{-1}\cdot\rho_{K_{h_{\epsilon}},\bar{\nabla}h_{\epsilon}(x_{\epsilon})}\left(\frac{M_{\epsilon}(u)}{\left|M_{\epsilon}(u)\right|}\right)\right)^{q-1}du$
$\displaystyle=\frac{\epsilon^{2-q}}{n}\int_{\mathbb{S}^{n-1}}\left|N_{\epsilon}y\right|^{q-1-n}\rho_{K_{h_{\epsilon}},\bar{\nabla}h_{\epsilon}(x_{\epsilon}))}^{q-1}(y)dy,$

where we apply the integration by substitution

$\begin{gathered}y=\frac{M_{\epsilon}(u)}{\left|M_{\epsilon}(u)\right|}=\frac{\epsilon
u^{\prime},u_{n}}{\left|M_{\epsilon}(u)\right|}\\\
u=\frac{M_{\epsilon}^{-1}y}{\left|M_{\epsilon}^{-1}y\right|}=\frac{\frac{1}{\epsilon}(y^{\prime},\epsilon
y_{n})}{\left|M_{\epsilon}^{-1}y\right|}.\end{gathered}$

then

$\displaystyle
u_{\epsilon}^{1-p}\widetilde{V}_{q-1}([u_{\epsilon}],\bar{\nabla}u_{\epsilon})\mbox{det}(\nabla^{2}u_{\epsilon}+u_{\epsilon})$
$\displaystyle=\epsilon^{p-q+\gamma-n+4}\left|N_{\epsilon}(x)\right|^{-\gamma-\alpha-
n-p}h_{\epsilon}^{1-p}(x_{\epsilon})\left|x^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}$
$\displaystyle\quad\frac{1}{n}\int_{\mathbb{S}^{n-1}}\left|N_{\epsilon}y\right|^{q-1-n}\rho_{K_{h_{\epsilon}},\bar{\nabla}h_{\epsilon}(x_{\epsilon})}^{q-1}(y)dy,$

Let $H_{\epsilon}(x):=\epsilon^{\frac{n-p-4-\gamma+q}{n-p+q-1}}u_{\epsilon},$
we have

$\displaystyle
H_{\epsilon}^{1-p}\widetilde{V}_{q-1}([H_{\epsilon}],\bar{\nabla}H_{\epsilon})\mbox{det}(\nabla^{2}H_{\epsilon}+H_{\epsilon}I)$
$\displaystyle=\left|N_{\epsilon}(x)\right|^{-\gamma-\alpha-
n-p}h_{\epsilon}^{1-p}(x_{\epsilon})\left|x^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}$
$\displaystyle\quad\frac{1}{n}\int_{\mathbb{S}^{n-1}}\left|N_{\epsilon}y\right|^{q-1-n}\rho_{K_{h_{\epsilon}},\bar{\nabla}h_{\epsilon}(x_{\epsilon})}^{q-1}(y)dy,$

∎

Since
$|y^{\prime}|\leqslant\left|N_{\epsilon}y\right|=\sqrt{|y^{\prime}|^{2}+\epsilon|y_{n}|^{2}}\leqslant
1,q\leqslant 1+n$ we have

$\displaystyle\frac{1}{n}\int_{\mathbb{S}^{n-1}}\left|N_{\epsilon}y\right|^{q-1-n}\rho_{K_{h_{\epsilon}},\bar{\nabla}h_{\epsilon}(x_{\epsilon})}^{q-1}(y)dy$
$\displaystyle\geqslant\frac{1}{n}\int_{\mathbb{S}^{n-1}}\rho_{K_{h_{\epsilon}},\bar{\nabla}h_{\epsilon}(x_{\epsilon})}^{q-1}(y)dy$
$\displaystyle\geqslant c.$

The second inequality is due to the uniform boundness of $h_{\epsilon},$
(3.2). Indeed, Since

$\int_{\mathbb{S}^{n-1}}\rho_{K_{h_{\epsilon}},\bar{\nabla}h_{\epsilon}(x_{\epsilon})}^{q-1}(y)dy=\frac{1}{2}\int_{S^{n-1}}X_{K_{h_{\epsilon}}}(\bar{\nabla}h_{\epsilon}(x_{\epsilon}),y)^{q-1}dy$

where the parallel $X$-ray of $K_{h_{\epsilon}}$ is the nonnegative function
on $\mathbb{R}^{n}\times S^{n-1}$ defined by

${K_{h_{\epsilon}}}(z,u)=|K_{h_{\epsilon}}\cap(z+\mathbb{R}u)|,\quad
z\in\mathbb{R}^{n},\quad u\in S^{n-1}.$

We can choose a Borel set $Z\subset\mathbb{S}^{n-1}$ satisfying $|Z|\geqslant
c,$ here, $c$ is a universal constant independent with $\epsilon,$ and
$\forall y\in Z$ we have

$X_{K_{h_{\epsilon}}}(\bar{\nabla}h_{\epsilon}(x_{\epsilon}),y)\geqslant c$

for some uniform constant $c$. Indeed, in dimension 2, since
$K_{h_{\epsilon}}$ is pinched between two bounded balls, $\forall z\in\partial
K_{h_{\epsilon}},$ there is a Borel set $Z\subset\mathbb{S}^{1}$ with
$\arcsin(\frac{\sqrt{3}c}{2C})\leqslant|Z|\leqslant\frac{2\pi}{3}$ such that
$y\in Z,X_{K_{h_{\epsilon}}}(z,y)\geqslant c,$ here, the constant $c(C)$ is
the radius of the inner(outer) ball of $K_{\epsilon}.$ The higher dimensional
case is analogous, just do a rotation.

Then combining with (3.2), we have $f_{\epsilon}(x)\geqslant
c_{1}\left|X^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}\left|N_{\epsilon}X\right|^{-\gamma-\alpha-
n-p}.$ Hence

$\displaystyle\int_{\mathbb{S}^{n-1}}f_{\epsilon}(x)dx$
$\displaystyle\geqslant
c_{1}\int_{\mathbb{S}^{n-1}}\left|x^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}\left|N_{\epsilon}x\right|^{-\gamma-\alpha-
n-p}dx$
$\displaystyle=2c_{1}\omega_{n-2}\int_{0}^{\frac{\pi}{2}}\sin\theta^{\alpha}\cos\theta^{\beta}\left(\sin\theta^{2}+\epsilon^{2}\cos\theta^{2}\right)^{-\frac{\gamma+\alpha+n+p}{2}}\sin\theta^{n-2}d\theta$
$\displaystyle\geqslant
c_{2}\int_{0}^{\frac{\pi}{4}}\theta^{\alpha+n-2}\left(\theta^{2}+\epsilon^{2}\right)^{-\frac{\gamma+\alpha+n+p}{2}}d\theta$
$\displaystyle+c_{2}\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}(\frac{\pi}{2}-\theta)^{\beta}\left(1+\epsilon^{2}(\frac{\pi}{2}-\theta)^{2}\right)^{-\frac{\gamma+\alpha+n+p}{2}}d\theta$
$\displaystyle\geqslant
c_{2}\int_{0}^{\frac{\pi}{4}}t^{\beta}\left(1+\epsilon^{2}t^{2}\right)^{-\frac{\gamma+\alpha+n+p}{2}}dt$
$\displaystyle\geqslant
c_{2}\left\\{\begin{array}[]{lll}\int_{0}^{\frac{\pi}{4}}t^{\beta}2^{-\frac{\gamma+\alpha+n+p}{2}}dt&\text{
when }0\leqslant\gamma+\alpha+n+p\\\
\int_{0}^{\frac{\pi}{4}}t^{\beta}dt&\text{ when
}0>\gamma+\alpha+n+p\end{array}\right.$ $\displaystyle\geqslant c,$

where the third inequality is due to the integration by substitution
$t=\frac{\pi}{2}-\theta,$ and we throw the first item; the fourth inequality
is because that $0<\epsilon<1/2,1\leqslant 1+\epsilon^{2}t^{2}<2.$
On the other hand, by (3.2), we have

$f_{\epsilon}(x)\leqslant
C\left|X^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}\left|N_{\epsilon}X\right|^{-\gamma-\alpha-
n-p}\frac{1}{n}\int_{S^{n-1}}\left|N_{\epsilon}y\right|^{q-1-n}\rho_{K_{h_{\epsilon}},\bar{\nabla}h_{\epsilon}(x_{\epsilon})}^{q-1}(y)dy,$

first, we estimate the integral, since
$|y^{\prime}|\leqslant\left|N_{\epsilon}y\right|=\sqrt{|y^{\prime}|^{2}+\epsilon|y_{n}|^{2}}\leqslant
1,$ and by (3.2) we have

$\displaystyle\int_{S^{n-1}}\left|N_{\epsilon}y\right|^{q-1-n}\rho_{K_{h_{\epsilon}},\bar{\nabla}h_{\epsilon}(x_{\epsilon}))}^{q-1}(y)dy$
$\displaystyle\leqslant C\int_{S^{n-1}}\left|N_{\epsilon}y\right|^{q-1-n}dy$
$\displaystyle\leqslant
C2\omega_{n-2}\int_{0}^{\frac{\pi}{2}}\left(\sin\theta^{2}+\epsilon^{2}\cos\theta^{2}\right)^{\frac{q-1-n}{2}}\sin\theta^{n-2}d\theta,$

Since

$\displaystyle\int_{0}^{\frac{\pi}{2}}\left(\sin\theta^{2}+\epsilon^{2}\cos\theta^{2}\right)^{\frac{q-1-n}{2}}\sin\theta^{n-2}d\theta$
$\displaystyle\leqslant
C\int_{0}^{\frac{\pi}{4}}\theta^{n-2}\left(\theta^{2}+\epsilon^{2}\right)^{(q-1-n)/2}d\theta$
$\displaystyle+C\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\left(1+\epsilon^{2}(\frac{\pi}{2}-\theta)^{2}\right)^{(q-1-n)/2}d\theta$
$\displaystyle\leqslant
C\left[\int_{0}^{\epsilon}\theta^{n-2}\epsilon^{q-1-n}d\theta+\int_{\epsilon}^{\frac{\pi}{4}}\theta^{q-3}d\theta+\frac{\pi}{4}\right]$
$\displaystyle\leqslant
C\left\\{\begin{array}[]{lll}\frac{\epsilon^{q-2}}{n-1}+\frac{\pi}{4}+C&\text{
when }q>2\\\ \frac{1}{n-1}+\frac{\pi}{4}+|\log\epsilon|&\text{ when }q=2\\\
c\epsilon^{q-2}&\text{ when }q<2\end{array}\right.$ $\displaystyle\leqslant
C\left\\{\begin{array}[]{lll}C&\text{ when }q>2\\\ |\log\epsilon|&\text{ when
}q=2\\\ \epsilon^{q-2}&\text{ when }q<2\end{array}\right.$

Since we have $2<q\leqslant n+1,$ then

$\displaystyle f_{\epsilon}(x)$ $\displaystyle\leqslant
C\left|x^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}\left|N_{\epsilon}X\right|^{-\gamma-\alpha-
n-p}$ $\displaystyle\leqslant
C\left\\{\begin{array}[]{lll}\left|x^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}&\text{
when }0>\gamma+\alpha+n+p\\\ \left|x^{\prime}\right|^{-\gamma-
p-n}\left|x_{n}\right|^{\beta}&\text{ when }0\leqslant\gamma+\alpha+n+p\\\
\end{array}\right.$

and $\|f_{\epsilon}\|_{L^{1}(\mathbb{S}^{n-1})}\geqslant c.$ When
$0\leqslant\gamma+\alpha+n+p,$ we need $-1<\gamma<-1-\frac{p}{n+q-1},$ then
$-\gamma-p-n$ satisfies the condition of $\alpha$ in Theorem 1.4. Hence,
$f_{\epsilon}$ satisfies (1.6). Now, we need to estimate the chord integral of
$K_{H_{\epsilon}}.$

######  Lemma 3.3.

We have the estimate for the $q-$th chord integral of $K_{H_{\epsilon}}$ as
follows:

$\displaystyle I_{q}(K_{H_{\epsilon}})$ $\displaystyle\leqslant
C\left\\{\begin{array}[]{lll}\epsilon^{2-\frac{3+\gamma}{n-p+q-1}(q+n-1)}&\text{
when }q>2\\\ \epsilon^{2-\frac{3+\gamma}{n-p+1}(n+1)}|\log\epsilon|&\text{
when }q=2\\\ \epsilon^{q-\frac{3+\gamma}{n-p+q-1}(q+n-1)}&\text{ when
}q<2\end{array}\right.$

###### Proof.

Since

$I_{q}(K_{H_{\epsilon}})=\frac{2q}{(n+q-1)\omega_{n}}\int_{\partial
K_{H_{\epsilon}}}H_{\epsilon}(\nu(z))\widetilde{V}_{q-1}(K_{H_{\epsilon}},\bar{\nabla}H_{\epsilon}(\nu(z)))\mathrm{d}\mathcal{H}^{n-1}(z),$

and recall the definition of $H_{\epsilon}$ (3.4) and by caculation above, we
have

$\displaystyle\widetilde{V}_{q-1}\left(K_{H_{\epsilon}},\bar{\nabla}H_{\epsilon}(x)\right)$
$\displaystyle=\epsilon^{\frac{n-p-4-\gamma+q}{n-p+q-1}(q-1)}\widetilde{V}_{q-1}\left(K_{u_{\epsilon}},\bar{\nabla}u_{\epsilon}(x)\right)$
$\displaystyle=\frac{\epsilon^{2-q+\frac{n-p-4-\gamma+q}{n-p+q-1}(q-1)}}{n}\int_{\mathbb{S}^{n-1}}\left|N_{\epsilon}y\right|^{q-1-n}\rho_{K_{h_{\epsilon}},\bar{\nabla}h_{\epsilon}(x_{\epsilon}))}^{q-1}(y)dy$
$\displaystyle\leqslant
C\epsilon^{2-q+\frac{n-p-4-\gamma+q}{n-p+q-1}(q-1)}\left\\{\begin{array}[]{lll}1&\text{
when }q>2\\\ |\log\epsilon|&\text{ when }q=2\\\ \epsilon^{q-2}&\text{ when
}q<2\end{array}\right.$

and combining the definition of $u_{\epsilon},$ we have

${\mbox{Vol}}(K_{H_{\epsilon}})=\epsilon^{\frac{n-p-4-\gamma+q}{n-p+q-1}n+1-n}{\mbox{Vol}}(K_{h_{\epsilon}})$

Since ${\mbox{Vol}}(K_{h_{\epsilon}})\leqslant C,$ then we have

${\mbox{Vol}}(K_{H_{\epsilon}})\leqslant
C\epsilon^{\frac{n-p-4-\gamma+q}{n-p+q-1}n+1-n}.$

Now, we can estimate the chord integral

$\displaystyle I_{q}(K_{H_{\epsilon}})$ $\displaystyle\leqslant
C\epsilon^{2-q+\frac{n-p-4-\gamma+q}{n-p+q-1}(q-1)}{\mbox{Vol}}(K_{H_{\epsilon}})\left\\{\begin{array}[]{lll}1&\text{
when }q>2\\\ |\log\epsilon|&\text{ when }q=2\\\ \epsilon^{q-2}&\text{ when
}q<2\end{array}\right.$ $\displaystyle\leqslant
C\left\\{\begin{array}[]{lll}\epsilon^{2-\frac{3+\gamma}{n-p+q-1}(q+n-1)}&\text{
when }q>2\\\ \epsilon^{2-\frac{3+\gamma}{n-p+1}(n+1)}|\log\epsilon|&\text{
when }q=2\\\ \epsilon^{q-\frac{3+\gamma}{n-p+q-1}(q+n-1)}&\text{ when
}q<2\end{array}\right.$

The proof of this lemma is completed. ∎

######  Remark 3.4.

When $q>2$, choose $-1<\gamma<-\frac{2p}{n+q-1}-1.$ Then

$I_{q}(K_{H_{\epsilon}})\leqslant\epsilon^{2-\frac{3+\gamma}{n-p+q-1}(q+n-1)}\rightarrow
0,\text{ as }\epsilon\rightarrow 0^{+}.$

When $q=2$, choose $-1<\gamma<-\frac{2p}{n+1}-1.$ Then

$I_{q}(K_{H_{\epsilon}})\leqslant\epsilon^{2-\frac{3+\gamma}{n-p+1}(n+1)}|\log\epsilon|\rightarrow
0,\text{ as }\epsilon\rightarrow 0^{+}.$

When $0<q<2$, $p\in(-\infty,-\frac{(2-q)(n+q-1)}{q}),$ choose
$-1<\gamma<-\frac{qp}{n+q-1}+q-3.$ Then

$I_{q}(K_{H_{\epsilon}})\leqslant\epsilon^{q-\frac{3+\gamma}{n-p+q-1}(q+n-1)}\rightarrow
0,\text{ as }\epsilon\rightarrow 0^{+}.$

##  4 Variational solution

This section is dedicated to solving the $L_{p}$ chord Minkowski problem in a
variational method. Let $C_{re}^{+}(\mathbb{S}^{n-1})$ denote the set of
rotationally symmetric, even, and positive continuous functions on
$\mathbb{S}^{n-1}.$ Let $f\in C_{re}^{+}(\mathbb{S}^{n-1})$ satisfies (1.6),
and real numbers $\alpha,\beta$ are as in Theorem 1.4. We consider the
maximization problem

$\sup_{h\in
C_{re}^{+}(\mathbb{S}^{n-1})}\left\\{\Phi_{p}(h):=\int_{\mathbb{S}^{n-1}}f(x)h(x)^{p},I_{q}(K_{h})=1\right\\}.$
(4.1)

######  Lemma 4.1.

Let $h_{i}\in C_{re}^{+}(\mathbb{S}^{n-1})$ be a maximizing sequence, then
there exists some uniform constant $C$ such that

$\frac{1}{C}\leqslant h_{i}\leqslant C\text{ as }i\rightarrow+\infty.$ (4.2)

###### Proof.

Since $h_{i}\in C_{re}^{+}(\mathbb{S}^{n-1})$ is a maximizing sequence; that
is $I_{q}(K_{i})=1$ and

$\lim_{i\rightarrow\infty}\Phi_{p}(h_{i})=\sup_{h\in
C_{re}^{+}(\mathbb{S}^{n-1})}\left\\{\Phi_{p}(h):I_{q}(K_{h})=1\right\\}$

where $K_{i}$ is the convex body uniquely determined by $h_{i}.$ By John’
lemma, we have $\frac{1}{n}E_{i}\subset K_{i}\subset E_{i}.$ y. Since $K_{i}$
is rotationally symmetric and even, $E_{i}$ is also rotationally symmetric and
even. Therefore, the centre of $E_{i}$ is at the origin and there exists a
unique rotationally symmetric matrix $A_{i}$ of the form

$A_{i}=\left(\begin{array}[]{cccc}r_{i}a_{i}^{\frac{1}{n}}&&&\\\ &\ddots&&\\\
&&r_{i}a_{i}^{\frac{1}{n}}&\\\
&&&r_{i}a_{i}^{\frac{1-n}{n}}\end{array}\right),\text{ where
}r_{i}>0,a_{i}>0\text{ are constants, }$

such that

$\displaystyle h_{E_{i}}(x)=\left|A_{i}x\right|\quad\text{ on
}\mathbb{S}^{n-1},$
$\displaystyle\rho_{E_{i}}(u)=\left|A_{i}^{-1}u\right|^{-1}\text{ on
}\mathbb{S}^{n-1}$ $\displaystyle{\mbox{Vol}}(E_{i})=\kappa_{n}r_{i}^{n}.$

Suppose to the contrary that

$\max_{x\in\mathbb{S}^{n-1}}h_{i}(x)\rightarrow+\infty\text{ or
}\min_{x\in\mathbb{S}^{n-1}}h_{i}(x)\rightarrow 0,$

as $i\rightarrow+\infty.$ Since we have

$\frac{1}{n}\left|A_{i}x\right|\leq
h_{i}(x)\leq\left|A_{i}X\right|,\quad\forall x\in\mathbb{S}^{n-1}$

which implies that

$r_{i}a_{i}^{\frac{1}{n}}\rightarrow+\infty\text{ or
}r_{i}a_{i}^{\frac{1}{n}}\rightarrow 0\text{ or
}r_{i}a_{i}^{\frac{1-n}{n}}\rightarrow+\infty\text{ or
}r_{i}a_{i}^{\frac{1-n}{n}}\rightarrow 0$

as $i\rightarrow+\infty,$ which must cause one of the four cases

$r_{i}\rightarrow+\infty\quad r_{i}\rightarrow 0\quad
a_{i}\rightarrow+\infty\quad a_{i}\rightarrow 0.$

Since

$h_{i}(x)\geq\frac{1}{n}h_{E_{i}}(X)=\frac{1}{n}\left|A_{i}x\right|\text{ on
}\mathbb{S}^{n-1},$

and $p<0$, from the assumption of $f$ we have

$\displaystyle\Phi_{p}(h_{i})$
$\displaystyle=\int_{\mathbb{S}^{n-1}}f(x)h_{i}(x)^{p}dx$
$\displaystyle\leqslant
n^{-p}\int_{\mathbb{S}^{n-1}}f(x)\left|A_{i}x\right|^{p}dx$
$\displaystyle\leqslant
Cn^{-p}\int_{\mathbb{S}^{n-1}}\left|x^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}\left|A_{i}x\right|^{p}dx$
$\displaystyle:=Cn^{-p}F(A_{i})$

we compute in the spherical coordinates as follows:

$\displaystyle F(A_{i})=$ $\displaystyle
r_{i}^{p}a_{i}^{\frac{p}{n}}\int_{\mathbb{S}^{n-1}}\left|x^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}\left(\left|x^{\prime}\right|^{2}+a_{i}^{-2}x_{n}^{2}\right)^{\frac{p}{2}}\mathrm{~{}d}x$
$\displaystyle=$ $\displaystyle
2r_{i}^{p}a_{i}^{\frac{p}{n}}\omega_{n-2}\int_{0}^{\frac{\pi}{2}}\sin^{\alpha}\theta\cos^{\beta}\theta\left(\sin^{2}\theta+a_{i}^{-2}\cos^{2}\theta\right)^{\frac{p}{2}}\sin^{n-2}\theta\mathrm{d}\theta$
$\displaystyle\leq$ $\displaystyle
Cr_{i}^{p}a_{i}^{\frac{p}{n}}\int_{0}^{\frac{\pi}{4}}\theta^{\alpha+n-2}\left(\theta^{2}+a_{i}^{-2}\right)^{\frac{p}{2}}\mathrm{~{}d}\theta$
$\displaystyle+Cr_{i}^{p}a_{i}^{\frac{p}{n}}\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\left(\frac{\pi}{2}-\theta\right)^{\beta}\left(1+a_{i}^{-2}\left(\frac{\pi}{2}-\theta\right)^{2}\right)^{\frac{p}{2}}\mathrm{~{}d}\theta.$

Case 1: When $a_{i}>3$. We can further estimate these following integrals:

$\displaystyle\int_{0}^{\frac{\pi}{4}}\left(\theta^{2}+a_{i}^{-2}\right)^{\frac{p}{2}}\theta^{\alpha+n-2}\mathrm{~{}d}\theta$
$\displaystyle\leq\int_{0}^{\frac{1}{a_{i}}}a_{i}^{-p}\theta^{\alpha+n-2}\mathrm{~{}d}\theta+\int_{\frac{1}{a_{i}}}^{\frac{\pi}{4}}\theta^{p}\theta^{\alpha+n-2}\mathrm{~{}d}\theta$
$\displaystyle=\frac{a_{i}^{-p-\alpha-n+1}}{\alpha+n-1}+\int_{\frac{1}{a_{i}}}^{\frac{\pi}{4}}\theta^{p+\alpha+n-2}\mathrm{~{}d}\theta$
$\displaystyle\leq C\begin{cases}1,&\text{ if }p+\alpha+n-1>0,\\\ \log
a_{i},&\text{ if }p+\alpha+n-1=0,\\\ a_{i}^{-p-\alpha-n+1},&\text{ if
}p+\alpha+n-1<0,\end{cases}$

and applying that $\beta>-1,$ we have

$\displaystyle\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\left(\frac{\pi}{2}-\theta\right)^{\beta}\left(1+a_{i}^{-2}\left(\frac{\pi}{2}-\theta\right)^{2}\right)^{\frac{p}{2}}\mathrm{~{}d}\theta$
$\displaystyle=\int_{0}^{\frac{\pi}{4}}t^{\beta}\left(1+a_{i}^{-2}t^{2}\right)^{\frac{p}{2}}\mathrm{~{}d}t$
$\displaystyle\leq C\int_{0}^{\frac{\pi}{4}}t^{\beta}\mathrm{d}t$
$\displaystyle\leq C.$

Then we obtain

$F(A_{i})\leq Cr_{i}^{p}a_{i}^{\frac{p}{n}}\begin{cases}1,&\text{ if
}p+\alpha+n-1>0,\\\ \log a_{i},&\text{ if }p+\alpha+n-1=0,\\\
a_{i}^{-p-\alpha-n+1},&\text{ if }p+\alpha+n-1<0.\end{cases}$

Cace 2: When $1/3\leqslant a_{i}\leqslant 3.$ Since
$\left|A_{i}x\right|=r_{i}a_{i}^{\frac{1}{n}}(\left|x^{\prime}\right|^{2}+a_{i}^{-2}x_{n}^{2})^{\frac{1}{2}},$
by the assumption of $a_{i}$ and $\alpha>1-n,\beta>-1,$ we have

$Cr_{i}\leqslant\left|A_{i}x\right|\leqslant\tilde{C}r_{i},\quad\forall
x\in\mathbb{S}^{n-1}.$

Then

$Cr_{i}^{p}\leqslant F(A_{i})\leqslant\tilde{C}r_{i}^{p},\quad\forall
x\in\mathbb{S}^{n-1}.$

Cace 3: When $a_{i}<1/3$, since we have

$F(A_{i})=2r_{i}^{p}a_{i}^{\frac{p}{n}-p}\omega_{n-2}\int_{0}^{\frac{\pi}{2}}\sin^{\alpha}\theta\cos^{\beta}\theta\left(a_{i}^{2}\sin^{2}\theta+\cos^{2}\theta\right)^{\frac{p}{2}}\sin^{n-2}\theta\mathrm{d}\theta.$

Then, applying $\alpha+n-1>0,$ we have

$\displaystyle F(A_{i})$ $\displaystyle\leq
Cr_{i}^{p}a_{i}^{\frac{p}{n}-p}\left[\int_{0}^{\frac{\pi}{4}}\theta^{\alpha+n-2}\left(a_{i}^{2}\theta^{2}+1\right)^{\frac{p}{2}}\mathrm{~{}d}\theta+\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\left(\frac{\pi}{2}-\theta\right)^{\beta}\left(a_{i}^{2}+\left(\frac{\pi}{2}-\theta\right)^{2}\right)^{\frac{p}{2}}\mathrm{~{}d}\theta\right]$
$\displaystyle\leq
Cr_{i}^{p}a_{i}^{\frac{p}{n}-p}\left[1+\int_{0}^{\frac{\pi}{4}}t^{\beta}\left(a_{i}^{2}+t^{2}\right)^{\frac{p}{2}}\mathrm{~{}d}t\right].$

And

$\displaystyle\int_{0}^{\frac{\pi}{4}}t^{\beta}\left(a_{i}^{2}+t^{2}\right)^{\frac{p}{2}}\mathrm{~{}d}t$
$\displaystyle\leq\int_{0}^{a_{i}}t^{\beta}a_{i}^{p}\mathrm{~{}d}t+\int_{a_{i}}^{\frac{\pi}{4}}t^{\beta}t^{p}\mathrm{~{}d}t$
$\displaystyle=\frac{a_{i}^{p+\beta+1}}{\beta+1}+\int_{a_{i}}^{\frac{\pi}{4}}t^{\beta+p}\mathrm{~{}d}t$
$\displaystyle\leq C\begin{cases}1,&\text{ if }\beta+p+1>0,\\\ |\log
a_{i}|,&\text{ if }\beta+p+1=0,\\\ a_{i}^{\beta+p+1,}&\text{ if
}\beta+p+1<0.\end{cases}$

Then we obtain

$F(A_{i})\leq Cr_{i}^{p}a_{i}^{\frac{p}{n}-p}\begin{cases}1,&\text{ if
}\beta+p+1>0,\\\ |\log a_{i}|,&\text{ if }\beta+p+1=0,\\\
a_{i}^{\beta+p+1},&\text{ if }\beta+p+1<0.\end{cases}$

######  Claim 4.2.

If one of the four caces $r_{i}\rightarrow+\infty\quad r_{i}\rightarrow 0\quad
a_{i}\rightarrow+\infty\quad a_{i}\rightarrow 0$ occurs, then
$F(A_{i})\rightarrow 0.$

###### Proof.

Since $I_{q}(K_{i})=1,$ and $\frac{1}{n}E_{i}\subset K_{i}\subset E_{i},$ We
have

$(\frac{1}{n})^{n+q-1}I_{q}(E_{i})=I_{q}(\frac{1}{n}E_{i})\leqslant
I_{q}(K_{i})\leqslant I_{q}(E_{i}).$

If we set $\Lambda=n^{q+n-1}$, then

$\Lambda^{-1}\leq I_{q}(E_{i})\leq\Lambda$

By 2.2, when $q\in(1,n+1)$ is not an integer, we have

$\displaystyle I_{q}(E_{i})$
$\displaystyle=\left\\{\begin{array}[]{lll}\left(r_{i}a_{i}^{\frac{1}{n}}\right)^{n+q-1}I_{q}\left(\left(r_{i}a_{i}^{\frac{1}{n}}\right)^{-1}E_{i}\right)&\text{
when }a_{i}>1\\\
\left(r_{i}a_{i}^{\frac{1-n}{n}}\right)^{n+q-1}I_{q}\left(\left(r_{i}a_{i}^{\frac{1-n}{n}}\right)^{-1}E_{i}\right)&\text{
when }a_{i}\leqslant 1\\\ \end{array}\right.$
$\displaystyle\leqslant\left\\{\begin{array}[]{lll}c_{q,m,n}\left(r_{i}a_{i}^{\frac{1}{n}}\right)^{n+q-1}a_{i}^{-2}&\text{
when }a_{i}>1\\\
c_{q,m,n}\left(r_{i}a_{i}^{\frac{1-n}{n}}\right)^{n+q-1}a_{i}^{n+q-2}&\text{
when }a_{i}\leqslant 1\\\ \end{array}\right.$ $\displaystyle\leqslant
c_{q,m,n}\left(r_{i}a_{i}^{\frac{1}{n}}\right)^{n+q-1}a_{i}^{-1}$

where the last inequality is due to $a_{i}>1,$ which leads to
$a_{i}^{-2}<a_{i}^{-1}.$
When $q\in(1,n+1)$ is an integer, we can choose a proper
$q^{\prime}\in(q,q+1),$ by 2.3, we have

$\displaystyle I_{q}(E_{i})$ $\displaystyle\leqslant
c(q^{\prime},q)V(E_{i})^{1-\frac{q-1}{q^{\prime}-1}}I_{q^{\prime}}(E_{i})^{\frac{q-1}{q^{\prime}-1}}$
$\displaystyle\leqslant\omega_{n}c(q^{\prime},q)c_{q^{\prime},q,n}\left(r_{i}a_{i}^{\frac{1}{n}}\right)^{n+q-1}a_{i}^{-1}.$

On the other hand,

$I_{q}(E_{i})=\frac{1}{n\omega_{n}}\int_{S^{n-1}}\int_{E_{i}|u^{\bot}}X_{E_{i}}(x,u)^{q}\mathrm{~{}d}x\mathrm{~{}d}u,\quad
q\geq 0.$

where $E_{i}|u^{\bot}$ denotes the projection of $E_{i}$ onto $u^{\bot}.$ For
$q>1,$ we have

$I_{q}(E_{i})\geqslant\frac{1}{n\omega_{n}}\int_{S^{n-1}}V(E_{i})^{q}V_{n-1}(E_{i}|u^{\bot})^{1-q}du$

Indeed, H$\ddot{o}$lder inequality gives

$\frac{1}{V_{n-1}(E_{i}|u^{\bot})}\int_{E_{i}|u^{\bot}}X_{E_{i}}(x,u)^{q}\mathrm{~{}d}x\geqslant\left(\frac{1}{V_{n-1}(E_{i}|u^{\bot})}\int_{E_{i}|u^{\bot}}X_{E_{i}}(x,u)\mathrm{~{}d}x\right)^{q}=\left(\frac{V(E_{i})}{V_{n-1}(E_{i}|u^{\bot})}\right)^{q}.$

Recall that $V(E_{i})=\kappa_{n}r_{i}^{n}.$
If $a_{i}>1$, $r_{i}a_{i}^{\frac{1}{n}}>r_{i}a_{i}^{\frac{1-n}{n}},$ it
follows that
$V_{n-1}(E_{i}|u^{\bot})\leqslant\kappa_{n-1}\left(r_{i}a_{i}^{\frac{1}{n}}\right)^{n-1}.$
Hence,

$\displaystyle I_{q}(E_{i})$
$\displaystyle\geqslant\frac{\kappa_{n-1}^{1-q}}{n\kappa_{n}^{1-q}}\left(r_{i}a_{i}^{\frac{1}{n}}\right)^{n-1}\left(r_{i}a_{i}^{\frac{1-n}{n}}\right)^{q}$
$\displaystyle=\frac{\kappa_{n-1}^{1-q}}{n\kappa_{n}^{1-q}}\left(r_{i}a_{i}^{\frac{1}{n}}\right)^{q+n-1}a_{i}^{-q}$

If $a\leqslant 1$, $r_{i}a_{i}^{\frac{1}{n}}\leqslant
r_{i}a_{i}^{\frac{1-n}{n}},$ it follows that
$V_{n-1}(E_{i}|u^{\bot})\leqslant\kappa_{n-1}\left(r_{i}a_{i}^{\frac{1}{n}}\right)^{n-2}r_{i}a_{i}^{\frac{1-n}{n}}.$
Hence,

$I_{q}(E_{i})\geqslant\frac{\kappa_{n-1}^{1-q}}{n\kappa_{n}^{1-q}}\left(r_{i}a_{i}^{\frac{1}{n}}\right)^{q+n-1}a_{i}^{-1}.$

In conlusion,

$\begin{gathered}\left(r_{i}a_{i}^{\frac{1}{n}}\right)^{q+n-1}a_{i}^{-q}\leqslant
C,\left(r_{i}a_{i}^{\frac{1}{n}}\right)^{q+n-1}a_{i}^{-1}\geqslant c\quad
a>1\\\
c\leqslant\left(r_{i}a_{i}^{\frac{1}{n}}\right)^{q+n-1}a_{i}^{-1}\leqslant
C\quad a\leqslant 1\end{gathered}$

where the constants $c,C$ only depend on $n,q.$ Therefore, we shall prove the
claim case by case as follows:
In case 1: $a>3,$ then $\left(r_{i}a_{i}^{\frac{1}{n}}\right)^{p}\leqslant
ca^{\frac{p}{n+q-1}}.$ Then we obtain

$F(A)\leq C\begin{cases}a^{\frac{p}{n+q-1}},&\text{ if }p+\alpha+n-1>0,\\\
a^{\frac{p}{n+q-1}}\log a,&\text{ if }p+\alpha+n-1=0,\\\
a^{\frac{p}{n+q-1}-p-\alpha-n+1},&\text{ if }p+\alpha+n-1<0.\end{cases}$

By the assumptions in Theorem 1.4, we observe that the power of $a$ is
negative. If one of $r_{i}\rightarrow+\infty\quad r_{i}\rightarrow 0\quad
a_{i}\rightarrow+\infty$ occurs, then $F(A_{i})\rightarrow 0.$
In case 2: If $1/3\leqslant a\leqslant 1,$ then
$ca^{\frac{1}{n+q-1}-\frac{1}{n}}\leqslant r_{i}\leqslant
Ca^{\frac{1}{n+q-1}-\frac{1}{n}};$ If $1\leqslant a\leqslant 3,$ then
$ca^{\frac{1}{n+q-1}-\frac{1}{n}}\leqslant r_{i}\leqslant
Ca^{\frac{q}{n+q-1}-\frac{1}{n}}.$ That is $c\leqslant r_{i}\leqslant C.$ Then
we obtain

$F(A)\leqslant\tilde{C}a^{\frac{p}{n+q-1}-\frac{p}{n}}\leqslant C.$

None of the four caces $r_{i}\rightarrow+\infty\quad r_{i}\rightarrow 0\quad
a_{i}\rightarrow+\infty\quad a_{i}\rightarrow 0$ can occur.
In case 3: $a<1/3,$ then
$\left(r_{i}a_{i}^{\frac{1}{n}}\right)^{p}a_{i}^{-p}\leqslant
ca^{\frac{p}{n+q-1}-p}.$ Then we obtain

$F(A)\leq C\begin{cases}a^{\frac{p}{n+q-1}-p},&\text{ if }\beta+p+1>0,\\\
a^{\frac{p}{n+q-1}-p}|\log a|,&\text{ if }\beta+p+1=0,\\\
a^{a^{\frac{p}{n+q-1}-p}+\beta+p+1},&\text{ if }\beta+p+1<0.\end{cases}$

By the assumptions in Theorem 1.4, we observe that the power of $a$ is
positive. If one of $r_{i}\rightarrow+\infty\quad r_{i}\rightarrow 0\quad
a_{i}\rightarrow 0$ occurs, then $F(A_{i})\rightarrow 0.$ ∎

This claim means that if one of the four cases
$r_{i}\rightarrow+\infty,r_{i}\rightarrow
0,a_{i}\rightarrow+\infty,a_{i}\rightarrow 0$ occurs, we have that

$\Phi_{p}(h_{i})\rightarrow 0,\quad\text{ as }\quad i\rightarrow+\infty$

However, taking $h\equiv r_{0}$ for some $r_{0}$ such that
$I_{q}(B_{r_{0}})=1.$ Indeed, since
$I_{q}(B_{r_{0}})=r_{0}^{n+q-1}I_{q}(B_{1}),$ we can choose
$r_{0}=\frac{\omega_{q}}{2^{q}\omega_{n}\omega_{n+q-1}}.$ Hence, we have

$\displaystyle\sup_{h\in
C_{re}^{+}(\mathbb{S}^{n-1})}\left\\{\Phi_{p}(h):I_{q}(K_{h})=1\right\\}$
$\displaystyle\geqslant$ $\displaystyle\int_{\mathbb{S}^{n-1}}fr_{0}^{p}$
(4.3) $\displaystyle\geqslant$ $\displaystyle
C_{1}\|f\|_{L^{1}\left(\mathbb{S}^{n-1}\right)}>0,$ (4.4)

which is a contradiction. Therefore, $\left\\{h_{i}\right\\}$ has uniformly
positive upper and lower bounds. ∎

######  Lemma 4.3.

The maximization problem has a solution $h.$

###### Proof.

Let $h_{i}\in C_{re}^{+}(\mathbb{S}^{n-1})$ be a maximizing sequence; that is
$I_{q}(K_{i})=1$ and

$\lim_{i\rightarrow\infty}\Phi_{p}(h_{i})=\sup_{h\in
C_{re}^{+}(\mathbb{S}^{n-1})}\left\\{\Phi_{p}(h):I_{q}(K_{h})=1\right\\}$

where $K_{i}$ is the convex body uniquely determined by $h_{i}.$ By 4.2 we
have $c\leqslant h_{i}\leqslant C$ as $i$ big enough. By the Blaschke
selection theorem, there is a subsequence of $\left\\{h_{i}\right\\}$ that
uniformly converges to some support function $h$ on $\mathbb{S}^{n-1}$. Note
that $h$ is also rotationally symmetric and even on $\mathbb{S}^{n-1}$,
satisfying (4.2), $I_{q}(K_{h})=1$, and

$\Phi_{p}(h)=\lim_{i\rightarrow+\infty}\Phi_{p}(h_{i})=\sup_{h\in
C_{re}^{+}(\mathbb{S}^{n-1})}\left\\{\Phi_{p}(h):I_{q}(K_{h})=1\right\\}$

Hence, $h$ is a solution to the maximization problem. ∎

######  Theorem 4.4.

Let $h$ be the maximizer obtained from lemma 4.3. Then $h$ is a generalized
solution of

$\mbox{det}(\nabla^{2}h+hI)=\frac{fh^{p-1}}{C\widetilde{V}_{q-1}([h],\bar{\nabla}h)},$

where $C=\frac{2q}{(n+q-1)\omega_{n}}\int fh^{p}.$

###### Proof.

Let $h$ be the maximizer obtained from lemma 4.3 , we have that $1/C\leqslant
h\leqslant C$ by (4.2). For any given rotationally symmetric and even $g\in
C(\mathbb{S}^{n-1}),$ let

$K_{t}=\left\\{x\in\mathbb{R}^{n}:x\cdot u\leqslant(h+tg)(u)\right\\}$

for sufficiently small $\delta>0$ such that $|t|<\delta$, we have $h+tg>0,$
and $h_{t}\in C_{re}^{+}(\mathbb{S}^{n-1}),$ where $h_{t}$ is the support
function of $K_{t}.$ Note that $h_{0}=h,K_{0}=K.$ Let
$\lambda(t)=I_{q}(K_{t})^{-\frac{1}{n+q-1}},$ then $I_{q}(\lambda(t)K_{t})=1,$
and
$\lambda^{\prime}(0)=-\frac{1}{n+q-1}\int_{\mathbb{S}^{n-1}}g(v)dF_{q}(K,v).$
Denote

$\Psi_{p}(t)=\Phi_{p}(\lambda(t)h_{t}).$

As $h$ is a maximizer, the function $t\mapsto\Psi_{p}(t)$ attains its maximum
at $t=0$. However, $h_{t}$ may not be differentiable at $t=0.$ Let

$\psi_{p}(t)=\Phi_{p}(\lambda(t)(h+tg)).$

Since $K_{t}$ is the Wulff shape of $h+tg$, we have $h_{t}\leqslant h+tg.$
Since $f$ nonnegative, $\lambda(t)>0,p<0$ and

$\lambda(t)h_{t}\leqslant\lambda(t)(h+tg),$

thus we have

$\Psi_{p}(0)\geqslant\Psi_{p}(t)\geqslant\psi_{p}(t).$

Since $\Psi_{p}(0)=\psi_{p}(0),$ $\psi_{p}(t)$ also attains its maximum at
$t=0.$ Hence we have

$\lim_{t_{k}\rightarrow
0^{+}}\frac{\psi_{p}(t_{k})-\psi_{p}(0)}{t_{k}}\leqslant 0,$

for any convergent subsequence $\\{t_{k}\\}.$ Therefore, we have

$\int_{S^{n-1}}ph(x)^{p-1}f(x)\left(\lambda^{\prime}(0)h+g\right)\leqslant 0,$

since $p<0$ and we can also replace $g$ by $-g,$ it follows that

$\int fh(x)^{p-1}g=\frac{1}{n+q-1}\int fh(x)^{p}\int gdF_{q}(K,\cdot).$

Since $g\in C(\mathbb{S}^{n-1})$ is arbitrary, we conclude that

$\mbox{det}(\nabla^{2}h+hI)=\frac{fh^{p-1}}{C\widetilde{V}_{q-1}([h],\bar{\nabla}h)},$

where $C=\frac{2q}{(n+q-1)\omega_{n}}\int fh^{p}.$ ∎

Now, we can prove Theorem 1.4. Let $h$ be the maximizer obtained from lemma
4.3, after a suitable scaling $\alpha h$ solves (1.5) with
$\alpha=C^{\frac{1}{n+q-p-1}}$. Denote $h_{0}=\alpha h,$ $K_{0}$ is its
corresponding convex body. We already know that $h_{0}$ is a generalized
solution to (1.5). Now we want to estimate the chord integral of $K_{0}.$
Since
$I_{q}(K_{0})=C^{\frac{n+q-1}{n+q-p-1}}I_{q}(K)=C^{\frac{n+q-1}{n+q-p-1}},$ we
only need to estimate the bound of $C$ from below. Applying (4.2), $h$ has
uniform bound from above. Hence, we have

$\displaystyle C$ $\displaystyle=\frac{2q}{(n+q-1)\omega_{n}}\int fh^{p}$
$\displaystyle\geqslant
c\frac{2q}{(n+q-1)\omega_{n}}\|f\|_{L^{1}(\mathbb{S}^{n-1})}$
$\displaystyle>0.$

The proof of Theorem 1.4 is completed.

##  5 Proof to Theorem 1.1

In this section, we aim to prove Theorem 1.1. It’s a result which follows
easily from Section 3 and Section 4.

###### Proof.

When $p<0$ and $2<q<n+1,$ for any given $\epsilon\in(0,1/2),$ let

$f_{\epsilon}(x):=h_{\epsilon}\left(x_{\epsilon}\right)^{1-p}\left|X^{\prime}\right|^{\alpha}\left|x_{n}\right|^{\beta}\left|N_{\epsilon}X\right|^{-\gamma-\alpha-
n-p}\frac{1}{n}\int_{S^{n-1}}\left|N_{\epsilon}y\right|^{q-1-n}\rho_{K_{h_{\epsilon}},\bar{\nabla}h_{\epsilon}(x_{\epsilon})}^{q-1}(y)dy,$
(5.1)

where $h_{\epsilon},x_{\epsilon},\alpha,\beta,\gamma,N_{\epsilon}$ are as in
Section 3. And we already know $H_{\epsilon}$ is a generalized solution to
equation (1.5) with $f$ replaced by $f_{\epsilon}.$ From the analysis of
$f_{\epsilon}$ in Section 3, we know $f_{\epsilon}$ satisfies the condition
(1.6). Hence, by Theorem 1.4, we obtain a variational solution $h_{0}$, which
is also a generalized solution to equation (1.5) with $f$ replaced by
$f_{\epsilon},$ and the $q$-th chord integral of its corresponding convex body
has a uniform positive bound from below. That is

$I_{q}(K_{h_{0}})\geqslant c>0,$

where $c$ depends only on $n,p,q,\alpha,\beta$ and $\gamma$ and is independent
with $\epsilon.$
However, by Remark 3.4, we can choose $\epsilon$ small enough such that

$I_{q}(K_{H_{\epsilon}})<\frac{c}{2},$

here $c$ is the same as above. Hence, $H_{\epsilon}$ and $h_{0}$ are different
solutions to equation (1.5).
Therefore, the proof of Theorem 1.1 is completed. ∎

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