Some extensions of the Pr\'ekopa-Leindler inequality using Borell's stochastic approach

TL;DR

This paper extends the Prékopa-Leindler inequality by developing an abstract form that encompasses known and new functional inequalities, utilizing Borell's stochastic approach to provide a unified proof framework.

Contribution

It introduces a generalized form of the Prékopa-Leindler inequality that includes new related inequalities, based on Borell's stochastic method.

Findings

Unified framework for multiple inequalities

Introduction of new inequalities in Euclidean spaces

Proofs based on Borell's stochastic approach

Abstract

We present an abstract form of the Pr\'ekopa-Leindler inequality that includes several known -and a few new- related functional inequalities on Euclidean spaces. The method of proof and also the formulation of the new inequalities are based on Christer Borell's stochastic approach to Brunn-Minkowski type inequalities.