31 citations · 50 across the 13 of their papers we have counts for
13 papers
A very short unified proof of Dar's conjecture and Log-Brunn--Minkowski in the plane
Dongmeng Xi
This short note gives a very short unified proof of both Dar's conjecture and the log-Brunn--Minkowski inequality in the plane. A planar -Brunn-Minkowski inequality for $p\in…
The many-body Blaschke-Santaló type inequality via optimal transport
Shibing Chen, Yuanyuan Li, Dongmeng Xi +1
Let be origin-symmetric measurable sets of finite volume such that \[ \sum_{1\le i<j\le k}\langle x_i,x_j\rangle\le \binom{k}{2}, \qquad \forall\…
The Mahler Conjecture in Three Dimensions
Shibing Chen, Yuanyuan Li, Dongmeng Xi +1
The Mahler conjecture dates back to 1938. This paper solves the conjecture for general convex bodies in three dimensions by developing a method called the shadow flow. The equality…
The Symmetric Mahler Inequality in Dimension Three via Admissible Shadow Systems
Shibing Chen, Yuanyuan Li, Dongmeng Xi +1
The three-dimensional symmetric Mahler inequality states that, for every origin-symmetric convex body \(K=-K\subset \mathbb{R}^3\), \[ \VP(K)= |K|\,|K^\circ|\geq \frac{32}{3}. \] I…
Chord Measures in Integral Geometry and Their Minkowski Problems
Erwin Lutwak, Dongmeng Xi, Deane Yang +1
To the families of geometric measures of convex bodies (the area measures of Aleksandrov-Fenchel-Jessen, the curvature measures of Federer, and the recently discovered dual curvatu…
General Higher Order Mean Zonoids
Dylan Langharst, Dongmeng Xi
In 1970, Schneider introduced the higher-order difference body and the associated Rogers-Shephard inequality. Recently, Haddad, Langharst, Putterman, Roysdon and Ye expanded the co…
The Reverse-log-Brunn-Minkowski inequality
Dongmeng Xi
Firstly, we propose our conjectured Reverse-log-Brunn-Minkowski inequality (RLBM). Secondly, we show that the (RLBM) conjecture is equivalent to the log-Brunn-Minkowski (LBM) conje…
The Chord Minkowski problem in a critical interval
Lujun Guo, Dongmeng Xi, Yiming Zhao
Chord measures and chord measures were recently introduced by Lutwak-Xi-Yang-Zhang by establishing a variational formula regarding a family of fundamental integral geometric…
Dual curvature measures for log-concave functions
Yong Huang, Jiaqian Liu, Dongmeng Xi +1
We introduce dual curvature measures for log-concave functions, which in the case of characteristic functions recover the dual curvature measures for convex bodies introduced by Hu…
On the sine polarity and the -sine Blaschke-Santaló inequality
Qingzhong Huang, Ai-Jun Li, Dongmeng Xi +1
This paper is dedicated to study the sine version of polar bodies and establish the -sine Blaschke-Santaló inequality for the -sine centroid body. The -sine centroid…
The case of equality in geometric instances of Barthe's reverse Brascamp-Lieb inequalit
Karoly J. Boroczky, Pavlos Kalantzopoulos, Dongmeng Xi
The works of Bennett, Carbery, Christ, Tao and of Valdimarsson have clarified when equality holds in the Brascamp-Lieb inequality. Here we characterize the case of equality in the…
The Minkowski problem in the Gaussian probability space
Yong Huang, Dongmeng Xi, Yiming Zhao
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,…
Affine Orlicz Pólya-Szegö principles and their equality cases
Youjiang Lin, Dongmeng Xi
The conjecture about the Orlicz Pólya-Szegö principle posed in [43] is proved. The cases of equality are characterized in the affine Orlicz Pólya-Szegö principle with respect to St…