Concentration of measure via approximated Brunn--Minkowski inequalities

TL;DR

This paper demonstrates that an approximate Brunn--Minkowski inequality with volume distortion leads to Gaussian concentration of measure, applicable even in discrete spaces.

Contribution

It introduces a novel link between approximate geometric inequalities and measure concentration, extending results to discrete settings.

Findings

Proves that approximate Brunn--Minkowski inequalities imply Gaussian concentration.

Establishes applicability to discrete spaces.

Provides a new geometric approach to measure concentration phenomena.

Abstract

We prove that an approximated version of the Brunn--Minkowski inequality with volume distortion coefficient implies a Gaussian concentration-of-measure phenomenon. Our main theorem is applicable to discrete spaces.