Transport-entropy forms of direct and Converseblaschke-Santal{\'o} inequalities

TL;DR

This paper introduces new transport-entropy and functional formulations of the Blaschke-Santal{ó} inequality and explores their implications for s-concave functions, symmetrized inequalities, and Mahler's conjecture.

Contribution

It provides novel direct and reverse inequalities for s-concave functions and establishes equivalences related to Mahler's conjecture via reinforced log-Sobolev inequalities.

Findings

New sharp symmetrized transport-entropy inequalities for spherically invariant measures

Equivalent formulation of Mahler's conjecture as a reinforced log-Sobolev inequality on the sphere

Extended Blaschke-Santal{ó} inequalities for s-concave functions

Abstract

We explore alternative functional or transport-entropy formulations of the Blaschke-Santal{\'o} inequality and of its conjectured counterpart due to Mahler. In particular, we obtain new direct and reverse Blaschke-Santal{\'o} inequalities for s-concave functions. We also obtain new sharp symmetrized transport-entropy inequalities for a large class of spherically invariant probability measures, including the uniform measure on the unit Euclidean sphere and generalized Cauchy and Barenblatt distributions. Finally, we show that the Mahler's conjecture is equivalent to some reinforced log-Sobolev type inequality on the sphere.