A sharp symmetrized form of Talagrand's transport-entropy inequality for the Gaussian measure

TL;DR

This paper introduces a precise form of a transport-entropy inequality for the Gaussian measure, enhancing Talagrand's inequality, with implications for measure concentration and dual formulations.

Contribution

It presents a sharp, symmetrized version of Talagrand's inequality for Gaussian measures, linking it to the functional Santaló inequality and exploring extensions.

Findings

Derived a sharp transport-entropy inequality for Gaussian measure

Connected the inequality to the functional Santaló inequality

Discussed extensions and implications for measure concentration

Abstract

This note presents a sharp transport-entropy inequality that improves on Talagrand's inequality for the Gaussian measure, arising as a dual formulation of the functional Santal\'o inequality. We also discuss some extensions and connections with concentration of measure.