Short probabilistic proof of the Brascamp-Lieb and Barthe theorems

TL;DR

This paper presents a concise probabilistic proof of the Brascamp-Lieb and Barthe theorems, demonstrating that Gaussian functions optimize certain inequalities, inspired by Borell's stochastic approach.

Contribution

It introduces a novel, short probabilistic proof for the Brascamp-Lieb and Barthe theorems, expanding the toolkit for understanding these inequalities.

Findings

Gaussian functions saturate the inequalities

Proof technique inspired by Borell's stochastic method

Applicable to both standard and reversed inequalities

Abstract

We give a short proof of the Brascamp-Lieb theorem, which asserts that a certain general form of Young's convolution inequality is saturated by Gaussian functions. The argument is inspired by Borell's stochastic proof of the Pr\'ekopa-Leindler inequality and applies also to the reversed Brascamp-Lieb inequality, due to Barthe.