On p-Brunn-Minkowski and Brascamp-Lieb inequalities

TL;DR

This paper establishes a connection between strong Brascamp--Lieb inequalities for symmetric log-concave measures and p-Brunn--Minkowski inequalities for level sets, leading to new results including the local log-Brunn--Minkowski inequality for L_q-balls.

Contribution

It introduces a novel equivalence between Brascamp--Lieb inequalities and p-Brunn--Minkowski inequalities, and proves the local log-Brunn--Minkowski inequality for all L_q-balls with q ≥ 1.

Findings

Proved equivalence between Brascamp--Lieb and p-Brunn--Minkowski inequalities.

Established new sufficient conditions for symmetric p-Brunn--Minkowski inequalities with p<1.

Extended the local log-Brunn--Minkowski inequality to all L_q-balls for q ≥ 1.

Abstract

We show that a strong version of the Brascamp--Lieb inequality for symmetric log-concave measure with $\alpha$-homogeneous potential $V$ is equivalent to a $p$-Brunn--Minkowski inequality for level sets of $V$ with some $p(\alpha,n)<0$. We establish links between several inequalities of this type on the sphere and the Euclidean space. Exploiting these observations, we prove new sufficient conditions for symmetric $p$-Brunn--Minkowski inequality with $p<1$. In particular, we prove the local log-Brunn--Minkowski for $L_q$-balls for all $q\geq 1$ in all dimensions, which was previously known only for $q\geq 2$.