arXiv subjects

Sudan Xing

Publications and source records attributed to Sudan Xing.

12 recordsLinked to original sources

arXiv2026

Fixed Point Rigidity of the Operator $\Gamma_p\Pi_p^\ast$ and the LYZ Conjecture

Motivated by the recent approach of Milman, Shabelman, and Yehudayoff \cite{MilmanShabelmanYehudayoff2025}, we establish, for $p>1$, a complete characterization of the fixed points of the composition of the $L_p$-centroid operator and the polar $L_p$-projection operator. More precisely, for $p>1$, we prove that if a convex body $K \in \mathcal{K}_o^n$ satisfies \[ \Gamma_p \Pi_p^* K = cK \] for some constant $c>0$, then $K$ must be an ellipsoid. Together with the result of case $p=1$, which was explicitly solved in the paper \cite{MilmanShabelmanYehudayoff2025}, this confirms the conjecture of Lutwak, Yang, and Zhang \cite{LutwakYangZhang2000} for $p\geq1$. Our approach combines variational techniques with a refined analysis of linear reflection shadow systems. We introduce a geometric framework, called the $L_p$-projection Rolodex, that represents the volume of the polar $L_p$-projection body in terms of weighted lower-dimensional sections. This representation yields a monotonicity property of the volume $\operatorname{Vol}_n(\Pi_p^*K_t)$ along linear reflection shadow systems $K_t$ and leads to a rigidity statement showing that the vanishing of the first variation forces constancy along the deformation. These results, together with known characterizations of equality in Steiner symmetrization, give the desired classification of fixed points.

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arXiv2025

New fiber and graph combinations of convex bodies

Three new combinations of convex bodies are introduced and studied: the $L_p$ fiber, $L_p$ chord and graph combinations. These combinations are defined in terms of the fibers and graphs of pairs of convex bodies, and each operation generalizes the classical Steiner symmetral, albeit in different ways. For the $L_p$ fiber and $L_p$ chord combinations, we derive Brunn--Minkowski-type inequalities and the corresponding Minkowski's first inequalities. We also prove that the general affine surface areas are concave (respectively, convex) with respect to the graph sum, thereby generalizing fundamental results of Ye (Indiana Univ. Math. J., 2014) on the monotonicity of the general affine surface areas under Steiner symmetrization. As an application, we deduce a corresponding Minkowski's first inequality for the $L_p$ affine surface area of a graph combination of convex bodies.

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arXiv2022

On Multiple $L_p$-curvilinear-Brunn-Minkowski inequalities

We construct the extension of the curvilinear summation for bounded Borel measurable sets to the $L_p$ space for multiple power parameter $\barα=(α_1, \cdots, α_{n+1})$ when $p>0$. Based on this $L_{p,\barα}$-curvilinear summation for sets and concept of {\it compression} of sets, the $L_{p,\barα}$-curvilinear-Brunn-Minkowski inequality for bounded Borel measurable sets and its normalized version are established. Furthermore, by utilizing the hypo-graphs for functions, we enact a brand new proof of $L_{p,\barα}$ Borell-Brascamp-Lieb inequality, as well as its normalized version, for functions containing the special case of $L_{p}$ Borell-Brascamp-Lieb inequality through the $L_{p,\barα}$-curvilinear-Brunn-Minkowski inequality for sets. Moreover, we propose the multiple power $L_{p,\barα}$-supremal-convolution for two functions together with its properties. Last but not least, we introduce the definition of the surface area originated from the variation formula of measure in terms of the $L_{p,\barα}$-curvilinear summation for sets as well as $L_{p,\barα}$-supremal-convolution for functions together with their corresponding Minkowski type inequalities and isoperimetric inequalities for $p\geq1,$ etc.

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arXiv2021

On the framework of $L_{p}$ summations for functions

We develop the framework of $L_p$ operations for functions by introducing two primary new types $L_{p,s}$ summations for $p>0$: the $L_{p,s}$ convolution sum and the $L_{p,s}$ Asplund sum for functions. The first type is defined as the linear summations of functions in terms of the $L_p$ coefficients ($C_{p,λ,t}$, $D_{p,λ,t}$), the so-called the $L_{p,s}$ supremal-convolution when $p\geq1$ and the $L_{p,s}$ inf-sup-convolution when $0<p<1$, respectively. The second type $L_{p,s}$ summation is created by the $L_p$ averages of bases for $s$-concave functions. We show that they are equivalent in the case $s=0$ (log-concave functions) and $p\geq1$. For the former type $L_{p,s}$ summation, we establish the corresponding $L_p$-Borell-Brascamp-Lieb inequalities for all $s\in[-\infty,\infty]$ and $p\geq1$. Furthermore, in summarizing the conditions for these new types of $L_p$-Borell-Brascamp-Lieb inequalities, we define a series of the $L_{p,s}$ concavity definitions for functions and measures. On the other hand, for the latter type $L_{p,s}$ Asplund summation, we discover the integral formula for $L_{p,s}$ mixed quermassintegral for functions via tackling the variation formula of quermassintegral of functions for $p\geq 1$.

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arXiv2021

On the Musielak-Orlicz-Gauss image problem

In the present paper we initiate the study of the Musielak-Orlicz-Brunn-Minkowski theory for convex bodies. In particular, we develop the Musielak-Orlicz-Gauss image problem aiming to characterize the Musielak-Orlicz-Gauss image measure of convex bodies. For a convex body $K$, its Musielak-Orlicz-Gauss image measure, denoted by $\widetilde{C}_Θ(K, \cdot)$, involves a triple $Θ=(G, Ψ, λ)$ where $G$ and $Ψ$ are two Musielak-Orlicz functions defined on $S^{n-1}\times (0, \infty)$ and $λ$ is a nonzero finite Lebesgue measure on the unit sphere $S^{n-1}$. Such a measure can be produced by a variational formula of $\widetilde{V}_{G, λ}(K)$ (the general dual volume of $K$ with respect to $λ$) under the perturbations of $K$ by the Musielak-Orlicz addition defined via the function $Ψ$. The Musielak-Orlicz-Gauss image problem contains many intensively studied Minkowski type problems and the recent Gauss image problem as its special cases. Under the condition that $G$ is decreasing on its second variable, the existence of solutions to this problem is established.

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arXiv2020

Geometry of log-concave functions: the $L_p$ Asplund sum and the $L_{p}$ Minkowski problem

The aim of this paper is to develop a basic framework of the $L_p$ theory for the geometry of log-concave functions, which can be viewed as a functional "lifting" of the $L_p$ Brunn-Minkowski theory for convex bodies. To fulfill this goal, by combining the $L_p$ Asplund sum of log-concave functions for all $p>1$ and the total mass, we obtain a Prékopa-Leindler type inequality and propose a definition for the first variation of the total mass in the $L_p$ setting. Based on these, we further establish an $L_p$ Minkowski type inequality related to the first variation of the total mass and derive a variational formula which motivates the definition of our $L_p$ surface area measure for log-concave functions. Consequently, the $L_p$ Minkowski problem for log-concave functions, which aims to characterize the $L_p$ surface area measure for log-concave functions, is introduced. The existence of solutions to the $L_p$ Minkowski problem for log-concave functions is obtained for $p>1$ under some mild conditions on the pre-given Borel measures.

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arXiv2020

On $L_p$-Brunn-Minkowski type and $L_p$-isoperimetric type inequalities for general measures

In 2011 Lutwak, Yang and Zhang extended the definition of the $L_p$-Minkowski convex combination ($p \geq 1$) introduced by Firey in the 1960s from convex bodies containing the origin in their interiors to all measurable subsets in $\mathbb{R}^n$, and as a consequence, extended the $L_p$-Brunn-Minkowski inequality ($L_p$-BMI) to the setting of all measurable sets. In this paper, we present a functional extension of their $L_p$-Minkowski convex combination---the $L_{p,s}$--supremal convolution and prove the $L_p$-Borell-Brascamp-Lieb type ($L_p$-BBL) inequalities. Based on the $L_p$-BBL type inequalities for functions, we extend the $L_p$-BMI for measurable sets to the class of Borel measures on $\mathbb{R}^n$ having $\left(\frac{1}{s}\right)$-concave densities, with $s \geq 0$; that is, we show that, for any pair of Borel sets $A,B \subset \mathbb{R}^n$, any $t \in [0,1]$ and $p\geq 1$, one has \[ μ((1-t) \cdot_p A +_p t \cdot_p B)^{\frac{p}{n+s}} \geq (1-t) μ(A)^{\frac{p}{n+s}} + t μ(B)^{\frac{p}{n+s}}, \] where $μ$ is a measure on $\mathbb{R}^n$ having a $\left(\frac{1}{s}\right)$-concave density for $0 \leq s < \infty$. Additionally, with the new defined $L_{p,s}$--supremal convolution for functions, we prove $L_p$-BMI for product measures with quasi-concave densities and for log-concave densities, $L_p$-Prékopa-Leindler type inequality ($L_p$-PLI) for product measures with quasi-concave densities, $L_p$-Minkowski's first inequality ($L_p$-MFI) and $L_p$ isoperimetric inequalities ($L_p$-ISMI) for general measures, etc. Finally a functional counterpart of the Gardner-Zvavitch conjecture is presented for the $p$-generalization.

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arXiv2019

The general dual-polar Orlicz-Minkowski problem

This paper gives a systematic study to the general dual-polar Orlicz-Minkowski problem (e.g., Problem \ref{general-dual-polar}). This problem involves the general dual volume $\widetilde{V}_G(\cdot)$ recently proposed in \cite{GHWXY, GHXY} in order to study the general dual Orlicz-Minkowski problem. As $\widetilde{V}_G(\cdot)$ extends the volume and the $q$th dual volume, the general dual-polar Orlicz-Minkowski problem is "polar" to the recently initiated general dual Orlicz-Minkowski problem in \cite{GHWXY, GHXY} and "dual" to the newly proposed polar Orlicz-Minkowski problem in \cite{LuoYeZhu}. The existence, continuity and uniqueness, if applicable, for the solutions to the general dual-polar Orlicz-Minkowski problem are established. Polytopal solutions and/or counterexamples to the general dual-polar Orlicz-Minkowski problem for discrete measures are also provided. Several variations of the general dual-polar Orlicz-Minkowski problem are discussed as well, in particular the one leading to the general Orlicz-Petty bodies.

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arXiv2018

General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski Problem II

The general dual volume $\dveV(K)$ and the general dual Orlicz curvature measure $\deV(K, \cdot)$ were recently introduced for functions $G: (0, \infty)\times \sphere\rightarrow (0, \infty)$ and convex bodies $K$ in $\R^n$ containing the origin in their interiors. We extend $\dveV(K)$ and $\deV(K, \cdot)$ to more general functions $G: [0, \infty)\times \sphere\rightarrow [0, \infty)$ and to compact convex sets $K$ containing the origin (but not necessarily in their interiors). Some basic properties of the general dual volume and of the dual Orlicz curvature measure, such as the continuous dependence on the underlying set, are provided. These are required to study a Minkowski-type problem for the dual Orlicz curvature measure. We mainly focus on the case when $G$ and $ψ$ are both increasing, thus complementing our previous work. The Minkowski problem asks to characterize Borel measures $μ$ on $\sphere$ for which there is a convex body $K$ in $\R^n$ containing the origin such that $μ$ equals $\deV(K, \cdot)$, up to a constant. A major step in the analysis concerns discrete measures $μ$, for which we prove the existence of convex polytopes containing the origin in their interiors solving the Minkowski problem. For general (not necessarily discrete) measures $μ$, we use an approximation argument. This approach is also applied to the case where $G$ is decreasing and $ψ$ is increasing, and hence augments our previous work. When the measures $μ$ are even, solutions that are origin-symmetric convex bodies are also provided under some mild conditions on $G$ and $ψ$. Our results generalize several previous works and provide more precise information about the solutions of the Minkowski problem when $μ$ is discrete or even.

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arXiv2018

General volumes in the Orlicz-Brunn-Minkowski theory and a related Minkowski Problem I

The general volume of a star body, a notion that includes the usual volume, the $q$th dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the $(p,q)$-dual curvature measures introduced recently by Lutwak, Yang, and Zhang. General variational formulas are established for the general volume of two types of Orlicz linear combinations. One of these is applied to the Minkowski problem for the new general dual Orlicz curvature measure, giving in particular a solution to the Minkowski problem posed by Lutwak, Yang, and Zhang for the $(p,q)$-dual curvature measures when $p>0$ and $q<0$. A dual Orlicz-Brunn-Minkowski inequality for general volumes is obtained, as well as dual Orlicz-Minkowski-type inequalities and uniqueness results for star bodies. Finally, a very general Minkowski-type inequality, involving two Orlicz functions, two convex bodies, and a star body, is proved, that includes as special cases several others in the literature, in particular one due to Lutwak, Yang, and Zhang for the $(p,q)$-mixed volume.

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arXiv2018

On the general dual Orlicz-Minkowski problem

For $K\subseteq \mathbb{R}^n$ a convex body with the origin $o$ in its interior, and $ϕ:\mathbb{R}^n\setminus\{o\}\rightarrow(0, \infty)$ a continuous function, define the general dual ($L_ϕ)$ Orlicz quermassintegral of $K$ by $$\mathcal{V}_ϕ(K)=\int_{\mathbb{R}^n \setminus K} ϕ(x)\,dx.$$ Under certain conditions on $ϕ$, we prove a variational formula for the general dual ($L_ϕ)$ Orlicz quermassintegral, which motivates the definition of $\widetilde{C}_{ϕ,\mathcal{V}}(K, \cdot)$, the general dual ($L_ϕ)$ Orlicz curvature measure of $K$. We pose the following general dual Orlicz-Minkowski problem: {\it Given a nonzero finite Borel measure $μ$ defined on $S^{n-1}$ and a continuous function $ϕ: \mathbb{R}^n\setminus\{o\}\rightarrow (0, \infty)$, can one find a constant $τ>0$ and a convex body $K$ (ideally, containing $o$ in its interior), such that,} $$μ=τ\widetilde{C}_{ϕ,\mathcal{V}}(K,\cdot)? $$ Based on the method of Lagrange multipliers and the established variational formula for the general dual ($L_ϕ)$ Orlicz quermassintegral, a solution to the general dual Orlicz-Minkowski problem is provided. In some special cases, the uniqueness of solutions is proved and the solution for $μ$ being a discrete measure is characterized.

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arXiv2017

The dual Orlicz-Minkowski problem

In this paper, the dual Orlicz curvature measure is proposed and its basic properties are provided. A variational formula for the dual Orlicz-quermassintegral is established in order to give a geometric interpretation of the dual Orlicz curvature measure. Based on the established variational formula, a solution to the dual Orlicz-Minkowski problem regarding the dual Orlicz curvature measure is provided.

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