On Gaussian Brunn-Minkowski inequalities

TL;DR

This paper explores Gaussian analogues of the classical Brunn-Minkowski inequality, providing streamlined proofs for related inequalities using semigroup methods, thereby advancing the understanding of geometric inequalities in Gaussian spaces.

Contribution

It introduces a unified semigroup approach to prove Gaussian versions of the Brunn-Minkowski, Ehrard, and Brascamp-Lieb inequalities, including their reverse forms.

Findings

Streamlined proofs of Gaussian Brunn-Minkowski inequalities

Semigroup methods applicable to Ehrard and Brascamp-Lieb inequalities

Unified approach simplifies understanding of geometric inequalities in Gaussian spaces

Abstract

In this paper, we are interested in Gaussian versions of the classical Brunn-Minkowski inequality. We prove in a streamlined way a semigroup version of the Ehrard inequality for $m$ Borel or convex sets based on a previous work by Borell. Our method also allows us to have semigroup proofs of the geometric Brascamp-Lieb inequality and of the reverse one which follow exactly the same lines.