On matrix-valued log-concavity and related Prekopa and Brascamp-Lieb inequalities

TL;DR

This paper introduces a novel approach to matrix-valued log-concavity, extending Prekopa's theorem, and establishes new inequalities including a Brascamp-Lieb variance inequality for matrix weights.

Contribution

It presents a self-contained method for extending Prekopa's theorem to matrix-valued functions and derives related inequalities, advancing the theoretical understanding of matrix log-concavity.

Findings

Extended Prekopa's theorem to matrix-valued functions

Established a Brascamp-Lieb variance inequality for matrix weights

Provided new inequalities related to matrix log-concavity

Abstract

We propose a new, self-contained, approach to H. Raufi's extension of Prekopa's theorem for matrix-valued log-concave functions. Along the way, new related inequalities are established, in particular a Brascamp-Lieb variance inequality for matrix weights.