The Existence of Solution to the Even Orlicz Chord Minkowski Problem

Qiuyue CHEN; Hailin JIN; Dandan LAI; Suwei LI

doi:10.1051/wujns/2025306529

Open Access

Issue		Wuhan Univ. J. Nat. Sci. Volume 30, Number 6, December 2025


Page(s)		529 - 534
DOI		https://doi.org/10.1051/wujns/2025306529
Published online		09 January 2026

Wuhan University Journal of Natural Sciences, 2025, Vol.30 No.6, 529-534

Mathematics

CLC number: O186

The Existence of Solution to the Even Orlicz Chord Minkowski Problem

偶Orlicz弦Minkowski问题解的存在性

Qiuyue CHEN (陈秋月)¹, Hailin JIN (金海林)¹^†, Dandan LAI (赖丹丹)² and Suwei LI (李苏薇)¹

¹ School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, Jiangsu, China
² Department of Mathematics, Shanghai University, Shanghai 200444, China

^† Corresponding author. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 16 May 2024

Abstract

Chord measures are newly discovered translation-invariant geometric measures of convex bodies in $R^{n}$ Mathematical equation by Lutwak-Xi-Yang-Zhang (Communications on Pure and Applied Mathematics, 2024), which is an extension of the surface area measure. The Minkowski problems for chord measures was considered by Lutwak-Xi-Yang-Zhang. In this paper, we use variational method to solve the even Orlicz chord Minkowski problem. The obtained results are an extension of the even Orlicz Minkowski problem from Haberl-Lutwak-Yang-Zhang (Advances in Mathematics, 2010 ).

摘要

2024年，Lutwak等人引入凸体的弦测度 (Communications on Pure and Applied Mathematics, 2024)。经典的表面积测度是一种特殊的弦测度。关于弦测度的Minkowski问题是凸几何中新的研究热点。本文采用变分方法研究Orlicz情形下偶弦测度的Minkowski问题解的存在性，所得到的结果是Haberl-Lutwak-Yang-Zhang (Advances in Mathematics, 2010) 关于偶表面积测度的Orlicz Minkowski问题的推广。

Key words: chord integral / chord measure / Minkowski problem

关键字 : 弦积分 / 弦测度 / Minkowski问题

Cite this article: CHEN Qiuyue, JIN Hailin, LAI Dandan, et al. The Existence of Solution to the Even Orlicz Chord Minkowski Problem[J]. Wuhan Univ J of Nat Sci, 2025, 30(6): 529-534.

Biography: CHEN Qiuyue, female, Master candidate, research direction: convex geometric analysis. E-mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

Foundation item: Supported by the National Natural Science Foundation of China (12071277, 12071334)

© Wuhan University 2025

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

The classical Brunn-Minkowski theory is the core of convex geometry analysis. The classical Minkowski problem asks for the existence, uniqueness, and regularity of a convex body whose surface area measure is equal to a pre-given Borel measure. Lutwak^[1-2] developed the classical Brunn-Minkowski theory into $L_{p}$ Mathematical equation -Brunn-Minkowski theory. After that, the $L_{p}$ Minkowski problem and related researches can be found in Refs.[3-14].

The $L_{p}$ Mathematical equation Minkowski problem has been extended to the even Orlicz Minkowski problem by Habrel et al^[15] with a series of papers on the Orlicz Minkowski theory ^[16-17]. The Orlicz Minkowski problem and related topics can be found in the Refs.[18-29].

Recently, Lutwak et al^[30] introduced a new family of geometric measures by studying a variational formula on the geometric invariants of convex body integrals, which called chord integrals. Accordingly, the $L_{p}$ Mathematical equation chord Minkowski problem was considered. Xi et al^[31] solved the $L_{p}$ chord Minkowski problem when $p > 1$ , $q > 1$ and the symmetric case of $0 < p < 1$ . Guo et al^[32] solved the $L_{p}$ chord Minkowski problem for $0 \leq p < 1$ without symmetric assumptions. Li^[33] solved the discrete $L_{p}$ Mathematical equation chord Minkowski problem in the condition of $p < 0$ and $q > 0$ , and as for general Borel measure, Li also gave a proof but need $- n < p < 0$ and $1 < q < n + 1$ . Hu et al ^[34] used flow methods to get regularity of the chord log-Minkowski problem of $p = 0$ . In addition, Hu et al^[35] also found the smooth origin-symmetric solution for $L_{p}$ Mathematical equation chord Minkowski problem in the case of ${p > 0, q > 3} ⋃$ ${- n < p < 0,3 < q < n + 1}$ by using same flow method^[34]. Zhao et al^[36] generalized the $L_{p}$ chord Minkowski problem and sloved the existence of smooth solutions to the Orlicz chord Minkowski problem.

In this paper, we use variational method to solve the even Orlicz chord Minkowski problem.

Let $K^{n}$ Mathematical equation be the collection of convex bodies (compact convex sets with nonempty interior) in $R^{n}$ . For $K \in K^{n}$ , the chord integral $I_{q} (K)$ of $K$ is defined as follows:

$I_{q} (K) = \int_{ℒ^{n}}^{} {| K ⋂ L |}^{q} d L, q \geq 0,$ Mathematical equation

where $| K ⋂ L |$ Mathematical equation denotes the length of the chord $| K ⋂ L |$ , and the integration is with respect to the Haar measure on the Grassmannian $ℒ^{n}$ of lines in $R^{n}$ .

Chord integrals contain volume $V (K)$ Mathematical equation and surface area $S (K)$ as two important special cases:

$I_{1} (K) = V (K), I_{0} (K) = \frac{ω_{n - 1}}{n ω_{n}} S (K), I_{n + 1} (K) = \frac{n + 1}{ω_{n}} V {(K)}^{2},$ Mathematical equation

where $ω_{n}$ Mathematical equation is the volume enclosed by the unit sphere $S^{n - 1}$ .

We can see in Ref.[15] that the differential of $I_{q} (K)$ Mathematical equation defines a finite Borel measure $F_{q} (K, \cdot)$ on $S^{n - 1}$ . Precisely, for convex bodies $K$ and $L$ in $R^{n}$ , we have

$\frac{d}{d t} |_{t = 0^{+}} I_{q} (K + t L) = \int_{S^{n - 1}}^{} h_{L} (v) d F_{q} (K, v), q \geq 0,$ Mathematical equation

where $F_{q} (K, \cdot)$ Mathematical equation is called the q-th chord measure of K and $h_{L}$ is the support function of $L$ . The cases of $q = 0,1$ of this formula are classical, which are the variational formulas of surface area and volume.

$F_{0} (K, \cdot) = \frac{(n - 1) ω_{n - 1}}{n ω_{n}} S_{n - 2} (K, \cdot), F_{1} (K, \cdot) = S_{n - 1} (K, \cdot),$ Mathematical equation

where $S_{n - 2} (K, \cdot)$ Mathematical equation and $S_{n - 1} (K, \cdot)$ are respecting the $(n - 2)$ -th order and $(n - 1)$ -th order area measure of $K$ .

The Orlicz chord Minkowski problem was stated in Ref.[36] by the following form:

The Orlicz chord Minkowski problem: Suppose $φ : (0, \infty) \to (0, \infty)$ Mathematical equation is a continuous decreasing function. If $μ$ is an finite Borel measure on $S^{n - 1}$ which is not concentrated on a great subsphere of $S^{n - 1}$ , what are the necessary and sufficient conditions on $μ$ , such that there exist a convex body $K \in K_{o}^{n}$ in $R^{n}$ Mathematical equation and a positive constant $c$ so that

$μ = c φ (h_{K}) d F_{q} (K, \cdot) ?$ Mathematical equation

Remark 1 When $φ (s) = s^{1 - p}$ Mathematical equation , the Orlicz chord Minkowski problem is reduce to the $L_{p}$ chord Minkowski problem^[30-31]. When $q = 1$ , the Orlicz chord Minkowski problem is reduced to the Orlicz Minkowski problem^[15].

In this paper, we consider the even Orlicz chord Minkowski problem. Concretely, we prove the following theorem.

Theorem 1 Let $q > 1$ Mathematical equation and $0 < α < 1$ . Suppose $φ : (0, \infty) \to (0, \infty)$ is a continuous decreasing function. If $μ$ is an even Borel measure on $S^{n - 1}$ that is not concentrated on a great subsphere of $S^{n - 1}$ , then there exists a symmetric, convex body $K_{0}$ such that

$c φ (h_{K_{0}}) d F_{q} (K_{0}, \cdot) = d μ,$ Mathematical equation

with $c = I_{q} (K_{0})^{\frac{α - (n + q - 1)}{n + q - 1}} .$ Mathematical equation

This paper is organized as follows. In Section 1, we introduce some basic facts about convex bodies. In Section 2, we give some lemmas needed for the proof of theorems in Section 3. In Section 3, we prove the main theorem.

1 Preliminaries

In this section, we will give some relative notations and facts about convex bodies. And for more details, see Refs.[37-39].

Let $R^{n}$ Mathematical equation be n-dimensional Euclidean space. The standard inner product in $R^{n}$ is denoted by $x \cdot y$ . We write $S^{n - 1} = {x \in R^{n} : x \cdot x = 1}$ for the boundary of the Euclidean unit ball $B$ in $R^{n}$ .

A convex body is a compact convex subset of $R^{n}$ Mathematical equation with non-empty interior. The set of convex bodies in $R^{n}$ containing the origin in their interiors is denoted by $K_{o}^{n}$ . The set of convex bodies in $R^{n}$ that are symmetric about the origin will be denoted by $K_{e}^{n}$ .

A compact, convex set $K \in R^{n}$ Mathematical equation is uniquely determined by its support function $h_{K} : R^{n} \to R$ , where $h_{K} (x) = m a x {x \cdot y : y \in K}$ , for each $x \in R^{n}$ . For example, the support function of the line segment $\bar{v}$ joining the points $\pm v \in R^{n}$ is given by

$h_{\bar{v}} (x) = | x \cdot v |, x \in R^{n} .$ Mathematical equation

It is trivial that for the support function of the dilate $c K = {c x : x \in K}$ Mathematical equation of a compact, convex $K$ , we have

$h_{c K} = c h_{K}, c > 0 .$ Mathematical equation (1)

Note that support functions are positively homogeneous of degree 1 and subadditive. It follows immediately from the definition of support functions that for compact, convex $K$ Mathematical equation , $L \subset R^{n}$ ,

$K \subseteq L \Leftrightarrow h_{K} \leq h_{L} .$ Mathematical equation (2)

Consequently, the support function of a body $K \in K_{o}^{n}$ Mathematical equation is bounded from above and below by positive reals.

The set of continuous functions on the sphere $S^{n - 1}$ Mathematical equation will be denoted by $C (S^{n - 1})$ and will always be viewed as equipped with the max-norm metric:

${‖ f - g ‖}_{\infty} = m a x_{u \in S^{n - 1}} | f (u) - g (u) |,$ Mathematical equation

for $f$ Mathematical equation , $g \in C (S^{n - 1})$ . The subspace of positive continuous functions will be denoted by $C^{+} (S^{n - 1})$ and the subspace of $C^{+} (S^{n - 1})$ consisting of only the even functions will be denoted by $C_{e}^{+} (S^{n - 1})$ .

For $h \in C (S^{n - 1})$ Mathematical equation , the Wulff-shape $[h]$ is a compact convex set defined by

$[h] = {x \in R^{n} : x \cdot v \leq h (v), \forall v \in S^{n - 1}} .$ Mathematical equation

Clearly, $h_{[h]} (v) \leq h (v)$ Mathematical equation .

The set $K_{o}^{n}$ Mathematical equation will be viewed as equipped with the Hausdorff metric. If a sequence $K_{i}$ of bodies in $K_{o}^{n}$ and a body $K \in K_{o}^{n}$ , we say that $\underset{i \to \infty}{l i m} K_{i} = K$ provided

${‖ h_{K_{i}} - h_{K} ‖}_{\infty} \to 0 .$ Mathematical equation

If $x \cdot u = h_{K} (u)$ Mathematical equation , a boundary point $x \in \partial K$ has $u \in S^{n - 1}$ as an outer normal. A boundary point is said to be singular if it has more than one unit normal vector. It is well known that the set of singular boundary points of a convex body has $ℋ^{n - 1}$ -measure equal to 0.

The inverse spherical image of a convex body $K$ Mathematical equation at $ω \subset S^{n - 1}$ which is a Borel set, denoted by $τ (K, ω)$ , is the set of all boundary points of $K$ which have an outer unit normal belonging to the set $ω$ . $S_{K}$ which is called the Aleksandrov-Fenchel-Jensen surface area measure of $K$ is defined by $S_{K} (ω) = ℋ^{n - 1} (τ (K, ω))$ Mathematical equation , for each Borel set $ω \subset S^{n - 1}$ , associated with each convex body $K \in K_{o}^{n}$ is a Borel measure on $S^{n - 1}$ .

2 Some Lemmas

This section mainly gives some lemmas needed for the proof in Section 3. First, we give the properties of a simple functional.

Lemma 1^[15] Suppose $φ : (0, \infty) \to (0, \infty)$ Mathematical equation is a continuous decreasing function. Define the function $ϕ : [0, \infty) \to$ $[0, \infty)$ by $ϕ (t) = \int_{0}^{t} \frac{1}{φ (s)} d s .$ For $c > 0$ and $0 < α < 1$ , we have

$\underset{t \to \infty}{l i m} \frac{ϕ (c t)}{t^{α}} = \infty$ Mathematical equation (3)

and

$\underset{t \to 0^{+}}{l i m} \frac{ϕ (t)}{t^{α}} = 0 .$ Mathematical equation (4)

Moreover, the derivative of $ϕ$ Mathematical equation is increasing and the function $ϕ$ is convex.

The following lemma is a slight variant of a standard result about differentiability under an integral sign.

Lemma 2^[15] Let $ϕ : (0, \infty) \to (0, \infty)$ Mathematical equation be continuously differentiable, $I \subset R$ be an open interval, and

$h : I \times S^{n - 1} \to (0, \infty), (t, u) \mapsto h (t, u)$ Mathematical equation

be a continuous function such that the partial derivative $\frac{\partial h}{\partial t} (t, u)$ Mathematical equation exists for all $(t, u) \in I \times S^{n - 1}$ . If $\frac{\partial h}{\partial t}$ is bounded and h is bounded from above and from below by positive numbers, then the function $H : I \to (0, \infty)$ defined by

$H (t) = \int_{S^{n - 1}}^{} (ϕ \circ h) (t, u) d μ (u),$ Mathematical equation

for $t \in I$ Mathematical equation , is differentiable on $I$ and

$H^{'} (t) = \int_{S^{n - 1}}^{} \frac{\partial (ϕ \circ h)}{\partial t} (t, u) d μ (u) .$ Mathematical equation

Moreover, if $\partial (ϕ \circ h) / \partial t$ Mathematical equation is continuous with respect to $t$ , then $H^{'}$ is continuous.

To solve the maximization problem, we use the variational formula in the following lemma.

Lemma 3^[30] Let $q > 0$ Mathematical equation . Suppose that $g : S^{n - 1} \to R$ is continuous and $h_{t} : S^{n - 1} \to (0, \infty)$ is a family of continuous functions given as follows:

$h_{t} = h_{0} + t g + o (t, \cdot),$ Mathematical equation

for each $t \in (- δ, δ)$ Mathematical equation with $δ > 0$ . Here, $ο (t, \cdot) \in C (S^{n - 1})$ and $ο (t, \cdot) / t$ tends to 0 uniformly on $S^{n - 1}$ as $t \to 0$ . Let $K_{t}$ be the Wulff-shape generated by $h_{t}$ and $K$ be the Wulff-shape generated by $h_{0}$ . Then,

$\frac{d}{d t} |_{t = 0} I_{q} (K_{t}) = \int_{S^{n - 1}}^{} g (v) d F_{q} (K, v) .$ Mathematical equation

3 The Even Orlicz Chord Minkowski Problem

Let $q \neq 0$ Mathematical equation and $0 < α < 1$ . Suppose $φ : (0, \infty) \to (0, \infty)$ is a continuous decreasing function. Define the function $ϕ : [0, \infty) \to [0, \infty)$ by $ϕ (t) = \int_{0}^{t} \frac{1}{φ (s)} d s .$ For each non-zero even finite Borel measure $μ$ on $S^{n - 1}$ Mathematical equation , define the functional $Φ_{ϕ, q} (h) : C^{+} (S^{n - 1}) \to R$ by

$Φ_{ϕ, q} (h) = \frac{n + q - 1}{α} I_{q} {([h])}^{\frac{α}{n + q - 1}} - \int_{S^{n - 1}}^{} (ϕ \circ h) d μ .$ Mathematical equation

The following theorem shows that if the maximization can be obtained, the even Orlicz chord Minkowski problem has a solution.

Theorem 2 Let $q > 0$ Mathematical equation and $μ$ be a nonzero even finite Borel measure on $S^{n - 1}$ that is not concentrated on a great subsphere of $S^{n - 1}$ . If the maximization problem

$s u p {Φ_{ϕ, q} (h) : h \in C^{+} (S^{n - 1})}$ Mathematical equation

has a solution $h_{0} \in C^{+} (S^{n - 1})$ Mathematical equation , then there exists $K_{0} \in K_{e}^{n}$ such that $c φ (h_{K_{0}}) d F_{q} (K_{0}, \cdot) = d μ .$

Proof Define $h_{t} = h_{0} + t g$ Mathematical equation for $g \in C (S^{n - 1})$ , $t \in (- δ, δ)$ for $δ > 0$ . For sufficiently small $| t |$ , the family $h_{t} \in C^{+} (S^{n - 1})$ . By Lemma 2 and Lemma 3, the function $t \to Φ_{ϕ, q} (h_{t})$ is differentiable at 0. By the fact that $h_{0}$ is a maximizer,

$\begin{array}{l} 0 = \frac{d}{d t} Φ_{ϕ, q} (h_{t}) |_{t = 0} \\ = I_{q} {([h_{0}])}^{\frac{α - (n + q - 1)}{n + q - 1}} \int_{S^{n - 1}}^{} g (v) d F_{q} ([h_{0}], v) \\ - \int_{S^{n - 1}}^{} \frac{1}{φ (h_{0})} g (v) d μ (v) . \end{array}$ Mathematical equation

Since $g \in C (S^{n - 1})$ Mathematical equation is arbitrary and using the fact that $h_{0} = h_{[h_{0}]}$ almost everywhere, for $F_{q} ([h_{0}], \cdot)$ , we have

$I_{q} {([h_{0}])}^{\frac{α - (n + q - 1)}{n + q - 1}} φ (h_{[h_{0}]}) d F_{q} ([h_{0}], v) = d μ (v) .$ Mathematical equation

Let $[h_{0}] = K_{0} \in K_{e}^{n}$ Mathematical equation , we have

$c φ (h_{K_{0}}) d F_{q} (K_{0}, v) = d μ (v),$ Mathematical equation

where $c = I_{q} (K_{0})^{\frac{α - (n + q - 1)}{n + q - 1}}$ Mathematical equation .

Next, we will prove the main theorem by showing that $Φ_{ϕ, q}$ Mathematical equation attains a maximum .

Theorem 3 Let $q > 0$ Mathematical equation and $0 < α < 1$ . Suppose $φ : (0, \infty) \to (0, \infty)$ is a continuous decreasing function. If $μ$ is an even finite Borel measure on $S^{n - 1}$ that is not concentrated on a great subsphere of $S^{n - 1}$ , then there exists a convex body $K_{0} \in K_{e}^{n}$ Mathematical equation such that

$c φ (h_{K_{0}}) d F_{q} (K_{0}, \cdot) = d μ$ Mathematical equation

with $c = I_{q} (K_{0})^{\frac{α - (n + q - 1)}{n + q - 1}}$ Mathematical equation .

Proof For $s > 0$ Mathematical equation , let $ψ_{s} : (0, \infty) \to R$ be defined by

$ψ_{s} (t) = (\frac{n + q - 1}{α} I_{q} {(B)}^{\frac{α}{n + q - 1}} - \frac{| μ | ϕ (s t)}{t^{α}}) t^{α} .$ Mathematical equation

From (3) we conclude that $\underset{t \to \infty}{l i m} ψ_{s} (t) = - \infty .$ Mathematical equation In particular, for each $s$ , there is a positive number $r_{s} > 0$ such that

$t > r_{s} \Rightarrow ψ_{s} (t) < 0 .$ Mathematical equation (5)

We use $B$ Mathematical equation to denote the centered unit ball in $R^{n}$ . For every $r > 0$ ,

$\begin{array}{l} Φ_{ϕ, q} (h_{r B}) = \frac{n + q - 1}{α} r^{α} I_{q} {(B)}^{\frac{α}{n + q - 1}} - ϕ (r) | μ | \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} \\ \begin{matrix} \end{matrix} = (\frac{n + q - 1}{α} I_{q} {(B)}^{\frac{α}{n + q - 1}} - \frac{| μ | ϕ (r)}{r^{α}}) r^{α} . \end{array}$ Mathematical equation

By the equation of (4), it follows that $Φ_{ϕ, q} (h_{r B})$ Mathematical equation is positive for small positive $r$ .

Hence, there exists $K \in K_{e}^{n}$ Mathematical equation such that

$Φ_{ϕ, q} (K) > 0 .$ Mathematical equation (6)

The Wulff-shape $[h]$ Mathematical equation associated with a given function $h \in C_{e}^{+} (S^{n - 1})$ is origin symmetric and has a support function $h_{[h]}$ with satisfies $0 < h_{[h]} \leq h$ . Since $ϕ$ is increasing and $I_{q} (h_{[h]}) = I_{q} (h)$ , we deduce $Φ_{ϕ, q} (h) \leq Φ_{ϕ, q} (h_{[h]}^{}) .$ Mathematical equation

Next, we will show that the search for a function at which $Φ_{ϕ, q}$ Mathematical equation attains a maximum can be further restricted to support functions of origin symmetric convex bodies contained in some ball of fixed radius. To this end, first note that the continuous function on $S^{n - 1}$ ,

$v \mapsto \int_{S^{n - 1}}^{} h_{\bar{v}} d μ,$ Mathematical equation

is positive since $μ$ Mathematical equation is not concentrated on a great subsphere. Thus, there exists a $s \in (0, \infty)$ such that

$\frac{1}{| μ |} \int_{S^{n - 1}}^{} h_{\bar{v}} d μ \geq s, f o r e v e r y^{} v \in S^{n - 1} .$ Mathematical equation (7)

Let $K \in K_{e}^{n}$ Mathematical equation and choose $v_{K} \in S^{n - 1}$ such that for a suitable real $r_{K} > 0$ the point $r_{K} v_{K}$ is an element of $K$ with maximal distance from the origin. Since $K$ is origin symmetric, the line segment with endpoints $\pm r_{K} v_{K}$ is contained in $K$ . From (3) and (2), we deduce $r_{K} h_{\bar{v} K} \leq h_{K}$ Mathematical equation . The monotonicity of $ϕ$ , Jensen's inequality, and (7) therefore yield

$\begin{array}{l} \int_{S^{n - 1}}^{} ϕ (h_{K}) d μ \geq \int_{S^{n - 1}}^{} ϕ (r_{K} h_{\bar{v} K}) d μ \\ \geq | μ | ϕ (\frac{1}{| μ |} \int_{S^{n - 1}}^{} r_{K} h_{\bar{v} K} d μ) \geq | μ | ϕ (s \cdot r_{K}) . \end{array}$ Mathematical equation

Since $K \subset r_{K} B$ Mathematical equation , the last inequality show that

$\begin{array}{l} Φ_{ϕ, q} (h_{K}) = \frac{n + q - 1}{α} I_{q} ([h_{K} {])}^{\frac{α}{n + q - 1}} - \int_{S^{n - 1}}^{} ϕ (h_{K}) d μ \\ \leq \frac{n + q - 1}{α} r_{k}^{α} I_{q} {(B)}^{\frac{α}{n + q - 1}} - | μ | ϕ (s \cdot r_{K}) \\ = (\frac{n + q - 1}{α} I_{q} {(B)}^{\frac{α}{n + q - 1}} - \frac{| μ | ϕ (s \cdot r_{K})}{r_{k}^{α}}) r_{k}^{α} \\ = ψ_{s} (r_{K}) . \end{array}$ Mathematical equation

From (6) we therefore conclude that there exists a real $r = r_{s} > 0$ Mathematical equation such that

$r_{K} > r \Rightarrow Φ_{ϕ, q} (K) < 0 .$ Mathematical equation (8)

It follows from (6) and (8) that in order to find a maximum of the functional $Φ_{ϕ, q}$ Mathematical equation on $C_{e}^{+} (S^{n - 1})$ , it is sufficient to search among support functions of members of the set

$ℱ = {K \in K_{e}^{n} : K \subset r B} .$ Mathematical equation

Let ${K_{i}}$ Mathematical equation be a maximizing sequence in $ℱ$ for $Φ_{ϕ, q}$ , i.e

$\underset{i \to \infty}{l i m} Φ_{ϕ, q} (h_{K_{i}}) = s u p {Φ_{ϕ, q} (h_{K}) : K \in ℱ} .$ Mathematical equation

Obviously, the sequence ${K_{i}}$ Mathematical equation is bounded. By the Blaschke's selection theorem, there exists a convergent subsequence, which we also denote by ${K_{i}}$ , with $\underset{i \to \infty}{l i m} K_{i} = K_{0}$ , where $K_{i}$ is origin symmetric. Hence, $K_{0}$ is origin symmetric. Let $h_{0} = h_{K_{o}}$ Mathematical equation and by the continuity of functional $Φ_{ϕ, q}$ , we deduce that

$\begin{array}{l} \frac{n + q - 1}{α} I_{q} {([h_{0}])}^{\frac{α}{n + q - 1}} \geq Φ_{ϕ, q} (h_{0}) = Φ_{ϕ, q} (h_{K_{0}}) = \underset{i \to \infty}{l i m} Φ_{ϕ, q} (h_{K_{i}}) \\ = s u p {Φ_{ϕ, q} (h_{K}) : K \in ℱ} > 0 . \end{array}$ Mathematical equation

Consequently, $K_{0}$ Mathematical equation has non-empty interior and thus $K_{0} \in K_{e}^{n}$ . Therefore, $h_{0}$ is a solution of the maximization problem for Theorem 2. This completes the proof.

Remark 2 When $φ (s) = s^{1 - p}$ Mathematical equation , Theorem 3 is reduce to the Theorem 4.2 of Ref.[31] in the case of $p > 1$ . When $q = 1$ , Theorem 1 is reduced to the Theorem 1 of Ref.[15].

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Xi D M, Jin H L, Leng G S. The Orlicz Brunn-Minkowski inequality[J]. Advances in Mathematics, 2014, 260: 350-374. [Google Scholar]
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Gardner R J. Geometry Tomography, Encyclopedia of Mathematics and Its Applications, 58[M]. Second Edition. Cambridge: Cambridge University Press, 2006. [Google Scholar]
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[R23] Ludwig M. General affine surface areas[J]. Advances in Mathematics, 2010, 224(6): 2346-2360. [Google Scholar]

[R24] Ludwig M, Reitzner M. A classification of SL(n) invariant valuations[J]. Annals of Mathematics, 2010, 172(2): 1219. [Google Scholar]

[R25] Xi D M, Jin H L, Leng G S. The Orlicz Brunn-Minkowski inequality[J]. Advances in Mathematics, 2014, 260: 350-374. [Google Scholar]

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[R27] Zou D, Xiong G. Orlicz-John ellipsoids[J]. Advances in Mathematics, 2014, 265: 132-168. [Google Scholar]

[R28] Zou D, Xiong G. The Orlicz Brunn-Minkowski inequality for the projection body[J]. The Journal of Geometric Analysis, 2020, 30(2): 2253-2272. [Google Scholar]

[R29] He M, Liu L J, Zeng H,The Orlicz Minkowski problem for logarithmic capacity[J]. Wuhan University Journal of Natural Sciences, 2025, 30(5): 471-478. [Google Scholar]

[R30] Lutwak E, Xi D, Yang D, et al. Chord measures in integral geometry and their Minkowski problems[J]. Communications on Pure and Applied Mathematics, 2024, 7: 77. [Google Scholar]

[R31] Xi D M, Yang D, Zhang G Y, et al. The L_p chord Minkowski problem[J]. Advanced Nonlinear Studies, 2023, 23: 20220041. [Google Scholar]

[R32] Guo L J, Xi D M, Zhao Y M, The L_p chord Minkowski problem[J]. Math Ann, 2024, 389: 3123-3162. [Google Scholar]

[R33] Li Y Y. The L_p chord Minkowski problem for negative p[J]. The Journal of Geometric Analysis, 2024, 34(3): 82. [Google Scholar]

[R34] Hu J R, Huang Y , Lu J. On the regularity of the chord log-Minkowski problem[EB/OL]. [2023-04-27].https://arxiv.org/abs/2304.14220. [Google Scholar]

[R35] Hu J R, Huang Y , Lu J, et al. The chord Gauss curvature flow and its L_p chord Minkowski problem[J]. Acta Mathematica Scientia, 2025, 45B(1): 161-179. [Google Scholar]

[R36] Zhao X, Zhao P B. Flow by Gauss curvature to the Orlicz chord Minkowski problem[J]. Annali Di Matematica Pura Ed Applicata, 2024, 203(5): 2405-2424. [Google Scholar]

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[R39] Thompson A C. Minkowski Geometry[M]. Cambridge: Cambridge University Press, 1996. [Google Scholar]