Rearrangement and Prekopa-Leindler type inequalities

TL;DR

This paper explores how functional rearrangements influence Prekopa-Leindler type inequalities, demonstrating that certain integral inequalities become tighter on specific rearranged sets and applying these results to various related inequalities.

Contribution

It introduces new insights into the interaction between rearrangements and Prekopa-Leindler inequalities, extending their applications to isoperimetric sets and Gaussian log-Sobolev inequalities.

Findings

Rearrangements tighten integral inequalities on isoperimetric sets.

Applications to Borell-Brascamp-Lieb and related inequalities are demonstrated.

Gaussian log-Sobolev inequality decreases on half-space rearrangement.

Abstract

We investigate the interactions of functional rearrangements with Prekopa-Leindler type inequalities. It is shown that that a general class of integral inequalities tighten on rearrangement to "isoperimetric" sets with respect to a relevant measure. Applications to the Borell-Brascamp-Lieb, Borell-Ehrhart, and the recent polar Prekopa-Leindler inequalities are demonstrated. It is also proven that an integrated form of the Gaussian log-Sobolev inequality decreases on half-space rearrangement.