Symmetrized Talagrand Inequalities on Euclidean Spaces

TL;DR

This paper explores symmetrized Talagrand inequalities in Euclidean spaces, providing new proofs, refined functional inequalities, and applications related to convex geometry and probability.

Contribution

It offers an alternative proof of Fathi's inequality, extends the inequality to Euclidean spaces, and derives refined functional inequalities and applications.

Findings

Alternative proof of Fathi's symmetrized Talagrand inequality on the real line

Refined functional inequalities under specific conditions

Connections established with convex geometry and applications

Abstract

In this paper, we study the symmetrized Talagrand inequality that was proved by Fathi and has a connection with the Blaschke-Santal\'{o} inequality in convex geometry. As corollaries of our results, we have several refined functional inequalities under some conditions. We also give an alternative proof of Fathi's symmetrized Talagrand inequality on the real line and some applications.