Rogers-Shephard inequality for log-concave functions

TL;DR

This paper extends the classical Rogers-Shephard inequalities to log-concave functions, providing new functional inequalities, characterizations of equality cases, and a novel approach using convolution bodies.

Contribution

It introduces a new method based on convolution bodies of log-concave functions to generalize Rogers-Shephard inequalities and characterizes equality cases.

Findings

Established new functional inequalities for log-concave functions.

Characterized all cases of equality in these inequalities.

Extended the notion of convolution bodies to functions, not just sets.

Abstract

In this paper we prove different functional inequalities extending the classical Rogers-Shephard inequalities for convex bodies. The original inequalities provide an optimal relation between the volume of a convex body and the volume of several symmetrizations of the body, such as, its difference body. We characterize the equality cases in all these inequalities. Our method is based on the extension of the notion of a convolution body of two convex sets to any pair of log-concave functions and the study of some geometrical properties of these new sets.