Some remarks on transportation cost and related inequalities

A Remark on discrete Brunn-Minkowski type inequalities via transportation of measure

We give an alternative proof for discrete Brunn-Minkowski type inequalities, recently obtained by Halikias, Klartag and the author. This proof also implies somewhat stronger weighted versions of these inequalities. Our approach generalizes ideas of Gozlan, Roberto, Samson and Tetali from the theory of measure transportation and provides new displacement convexity of entropy type inequalities for the lattice point enumerator.

马尔可夫核的输运不等式及其应用

Transportation inequalities for Markov kernels and their applications

本文研究度量空间上马尔可夫核的输运不等式，证明压缩不等式与反向Poincaré、反向对数Sobolev及Wang Harnack不等式等价，并应用于马尔可夫过程收敛和热核测度拟不变性。

A remark on perimeter-diameter and perimeter-circumradius inequalities under lattice constraints

In this note, we study several inequalities involving geometric functionals for lattice point-free planar convex sets. We focus on the previously not addressed cases perimeter--diameter and perimeter--circumradius.

Mean width inequalities of sections and projections for isotropic measures

In this paper, we establish mean width inequalities of sections and projections of convex bodies for isotropic measures with complete equality conditions, which extends the recent work of Alonso-Gutiérrez and Brazitikos. Different from their approach, our proof is based on the approach developed by Lutwak, Yang and Zhang, by using the Ball-Barthe inequality, the mass transportation, and the isotropic embedding.

An overview of L1 Optimal Transportation on metric measure spaces

The scope of this note is to make a self-contained survey of the recent developments and achievements of the theory of L1-Optimal Transportation on metric measure spaces. Among the results proved in the recent papers [20, 21] where the author, together with A. Mondino, proved a series of sharp (and in some cases rigid) geometric and functional inequalities in the setting of metric measure spaces enjoying a weak form of Ricci curvature lower bound, we review the proof of the Lévy-Gromov isoperimetric inequality.

Eigenvalue distribution of optimal transportation

We investigate the Brenier map $\nabla Φ$ between the uniform measures on two convex domains in $\mathbb{R}^n$ or more generally, between two log-concave probability measures on $\mathbb{R}^n$. We show that the eigenvalues of the Hessian matrix $D^2 Φ$ exhibit remarkable concentration properties on a multiplicative scale, regardless of the choice of the two measures or the dimension $n$.