Concentration of measures supported on the cube

TL;DR

This paper establishes a log-Sobolev inequality for a class of high-dimensional, convex, log-concave measures supported on the unit cube, using transportation-cost inequalities.

Contribution

It introduces a new log-Sobolev inequality for convex, non-strictly convex measures on the cube, expanding understanding of measure concentration in high dimensions.

Findings

Proves a log-Sobolev inequality for measures supported on the cube

Uses transportation-cost inequalities to derive results

Applies to measures with convex densities and bounded second derivatives

Abstract

We prove a log-Sobolev inequality for a certain class of log-concave measures in high dimension. These are the probability measures supported on the unit cube in R^n whose density takes the form exp(-H) where the function H is assumed to be convex (but not strictly convex) with bounded pure second derivatives. Our argument relies on a transportation-cost inequality a la Talagrand.