Transport inequalities for log-concave measures, quantitative forms and applications

TL;DR

This paper reviews techniques using monotone mass transport to derive transport inequalities for log-concave measures, explores their quantitative forms, and applies them to the Brascamp-Lieb variance inequality.

Contribution

It introduces simple transport-based methods for establishing inequalities for log-concave measures and discusses their quantitative aspects and applications.

Findings

Transport inequalities derived for log-concave measures

Quantitative forms of these inequalities are discussed

Applications to Brascamp-Lieb variance inequality demonstrated

Abstract

We review some simple techniques based on monotone mass transport that allow to obtain transport-type inequalities for any log-concave probability measure, and for more general measures as well. We discuss quantitative forms of these inequalities, with application to the Brascamp-Lieb variance inequality.