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

TL;DR

This paper presents an alternative proof for discrete Brunn-Minkowski inequalities using measure transportation, leading to stronger weighted versions and new displacement convexity inequalities for lattice point enumeration.

Contribution

It introduces a novel proof approach based on measure transportation, extending existing inequalities and establishing new convexity results for lattice point counts.

Findings

Alternative proof for discrete Brunn-Minkowski inequalities

Stronger weighted versions of these inequalities

New displacement convexity inequalities for lattice point enumeration

Abstract

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.