Mass transport and variants of the logarithmic Sobolev inequality

TL;DR

This paper advances the understanding of mass transport methods to establish modified log-Sobolev and isoperimetric inequalities, providing new conditions and unifying existing criteria, including in Riemannian contexts.

Contribution

It introduces novel sufficient conditions for functional inequalities using optimal transportation, applicable to non-convex measures and extended to Riemannian manifolds.

Findings

New criteria for modified log-Sobolev inequalities

Unified transportation-based proofs of classical results

Extension of inequalities to Riemannian settings

Abstract

We develop the optimal transportation approach to modified log-Sobolev inequalities and to isoperimetric inequalities. Various sufficient conditions for such inequalities are given. Some of them are new even in the classical log-Sobolev case. The idea behind many of these conditions is that measures with a non-convex potential may enjoy such functional inequalities provided they have a strong integrability property that balances the lack of convexity. In addition, several known criteria are recovered in a simple unified way by transportation methods and generalized to the Riemannian setting.