Transport proofs of some functional inverse Santal{\'o} inequalities

TL;DR

This paper provides a new, simplified proof connecting functional inverse-Santal{ó} inequalities to Entropy-Transport inequalities, and extends these results to one-dimensional and unconditional n-dimensional cases using transport methods.

Contribution

It introduces a straightforward proof linking inverse-Santal{ó} inequalities to Entropy-Transport inequalities and applies transport techniques to prove these inequalities in various dimensions.

Findings

Established a simple proof of the connection between inverse-Santal{ó} and Entropy-Transport inequalities.

Proved sharp Entropy-Transport inequalities in dimension 1.

Revisited and extended the proof of inverse-Santal{ó} inequalities in n-dimensional unconditional cases.

Abstract

In this paper, we present a simple proof of a recent result of the second author which establishes that functional inverse-Santal{\'o} inequalities follow from Entropy-Transport inequalities. Then, using transport arguments together with elementary correlation inequalities, we prove these sharp Entropy-Transport inequalities in dimension 1. We also revisit the proof of the functional inverse-Santal{\'o} inequalities in the n dimensional unconditional case using these ideas.