On the Log-Sobolev Constant of Log-Concave Vectors

TL;DR

This paper explores the relationship between log-Sobolev inequalities and subgaussian tails in log-concave vectors, improving bounds and examining specific cases to deepen understanding of their interplay.

Contribution

It provides the best dimension-dependent bounds on the log-Sobolev constant for subgaussian log-concave measures and investigates the reverse implication under log-concavity.

Findings

Improved dimension-dependent bounds on log-Sobolev constants.

Established conditions under which subgaussian tails imply log-Sobolev inequalities.

Analyzed special cases to clarify the relationship between tail behavior and functional inequalities.

Abstract

It is well known that if a random vector satisfies a log-Sobolev inequality, all of its marginals have subgaussian tails. In the spirit of the KLS conjecture, we investigate whether this implication can be reversed under a log-concavity assumption. In the general setting, we improve on a result of Bobkov, establishing the best dimension dependent bound on the log-Sobolev constant of subgaussian log-concave measures, and we investigate some special cases.