# Dongmeng XiShanghai University | SHU · College of Sciences

Dongmeng Xi

Doctor of Philosophy

## About

20

Publications

1,168

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

361

Citations

Introduction

Convex geometric analysis

**Skills and Expertise**

## Publications

Publications (20)

This paper is dedicated to study the sine version of polar bodies and establish the $L_p$-sine Blaschke-Santal\'{o} inequality for the $L_p$-sine centroid body. The $L_p$-sine centroid body $\Lambda_p K$ for a star body $K\subset\mathbb{R}^n$ is a convex body based on the $L_p$-sine transform, and its associated Blaschke-Santal\'{o} inequality prov...

This paper is dedicated to study the sine version of polar bodies and establish the Lp-sine Blaschke-Santaló inequality for the Lp-sine centroid body.
The Lp-sine centroid body ΛpK for a star body K⊂Rn is a convex body based on the Lp-sine transform, and its associated Blaschke-Santaló inequality provides an upper bound for the volume of Λp∘K, the...

The original goal of this paper is to extend the affine isoperimetric inequality and Steiner type inequality of Orlicz projection bodies (which originated to Lutwak, Yang, and Zhang [32]), from convex bodies to Lipschitz star bodies (whose radial functions are locally Lipschitz).
In order to achieve it, we investigate the graph functions of the giv...

From a convex geometry viewpoint, we proved the L p L_p Brunn-Minkowski inequalities for q q -th dual quermassintegrals, when p ≥ q p\geq q .
Based on these inequalities, we obtain relevant uniqueness results of the ( p , q ) (p,q) -th dual curvature measures (up to a dilation when p = q p=q ). As a special case q = 0 q=0 , we obtain the uniqueness...

The Minkowski problem in Gaussian probability space is studied in this paper. In addition to providing an existence result on a Gaussian-volume-normalized version of this problem, the main goal of the current work is to provide uniqueness and existence results on the Gaussian Minkowski problem (with no normalization required).

General affine invariances related to Mahler volume are introduced. We establish their affine isoperimetric inequalities by using a symmetrization scheme that involves a total of $2n$ elaborately chosen Steiner symmetrizations at a time. The necessity of this scheme, as opposed to the usual Steiner symmetrization, is demonstrated with an example (s...

A. R. Martínez Fernández obtained upper bounds for quermassintegrals of the p -inner parallel bodies: an extension of the classical inner parallel body to the $L_p$ -Brunn-Minkowski theory. In this paper, we establish (sharp) upper and lower bounds for quermassintegrals of p -inner parallel bodies. Moreover, the sufficient and necessary conditions...

The Minkowski problem in Gaussian probability space is studied in this paper. In addition to providing an existence result on a Gaussian-volume-normalized version of this problem, the main goal of the current work is to provide uniqueness and existence results on the Gaussian Minkowski problem (with no normalization required).

The conjecture about the Orlicz P\'olya-Szeg\"o principle posed in [43] is proved. The cases of equality are characterized in the affine Orlicz P\'olya-Szeg\"o principle with respect to Steiner symmetrization and Schwarz spherical symmetrization.

The Orlicz Minkowski problem is a generalization of the Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_p$$\end{document} Minkowski problem. For a class of appropri...

Corresponding to the Legendre ellipsoid and the LYZ ellipsoid, two new sine ellipsoids are introduced in this paper. These four ellipsoids are closely related in the Pythagorean relation and duality. Several volume inequalities and the valuation properties are obtained for two new ellipsoids.

The conjecture about the Orlicz Pólya–Szegö principle posed in [42] is proved. The cases of equality are characterized in the affine Orlicz Pólya–Szegö principle with respect to Steiner symmetrization and Schwarz spherical symmetrization.

In this paper, we demonstrate the existence part of the discrete Orlicz Minkowski problem, which is a non-trivial extension of the discrete L p Minkowski problem for 0 < p < 1.

Let . Sharp volume inequalities for k-dimensional sections of Wulff shapes and dual inequalities for projections are established. As their applications, several special Wulff shapes are investigated.

Let \(1\le k\le n\). Sharp upper and lower bounds for the volume of k-dimensional projections (or sections) of \(L_p\)-zonoids (or their polars) with even isotropic generating measures are established. As special cases, sharp volume inequalities for k-dimensional sections and projections of \(\ell _p^n\)-balls are recovered. The necessary condition...

The cosine transform on Grassmann manifolds naturally induces finite dimensional Banach norms whose unit balls are origin-symmetric convex bodies in . Reverse isoperimetric type volume inequalities for these bodies are established, which extend results from the sphere to Grassmann manifolds.

In 1999, Dar conjectured that there is a stronger version of the celebrated Brunn-Minkowski inequality. However, as pointed out by Campi, Gardner, and Gronchi in 2011, this problem seems to be open even for planar o-symmetric convex bodies. In this paper, we give a positive answer to Dar's conjecture for all planar convex bodies. We also give the e...

The Orlicz Brunn–Minkowski theory originated with the work of Lutwak, Yang, and Zhang in 2010. In this paper, we first introduce the Orlicz addition of convex bodies containing the origin in their interiors, and then extend the LpLp Brunn–Minkowski inequality to the Orlicz Brunn–Minkowski inequality. Furthermore, we extend the LpLp Minkowski mixed...

In this paper, the Lp (p ≥ 1) mean zonoid of a convex body K is given, and we show that it is the Lp centroid body of radial (n + p)th mean body of K up to a dilation. We also establish some affine inequalities of these bodies by proving that the volume of the new bodies is decreasing under Steiner symmetrization.