Some functional forms of Blaschke-Santal\'o inequality

TL;DR

This paper develops new functional inequalities related to the Blaschke-Santaló inequality, extending classical results to non-symmetric convex bodies and log-concave functions, with simplified proofs and equality characterizations.

Contribution

It introduces novel functional versions of the Blaschke-Santaló inequality applicable to non-symmetric cases and provides simplified proofs and equality conditions.

Findings

New functional inequalities for volume products of convex bodies.

Extensions to log-concave functions and Legendre transforms.

Simplified proof of the equality case.

Abstract

We establish new functional versions of the Blaschke-Santal\'o inequality on the volume product of a convex body which generalize to the non-symmetric setting an inequality of K. Ball and we give a simple proof of the case of equality. As a corollary, we get some inequalities for $\log$-concave functions and Legendre transforms which extend the recent result of Artstein, Klartag and Milman, with its equality case.