A Rogers--Brascamp--Lieb--Luttinger inequality in the space of matrices

TL;DR

This paper extends a classical inequality to the space of matrices, using a new fiber-symmetrization technique, and demonstrates its applications to quasi-concave functions.

Contribution

It introduces a Rogers--Brascamp--Lieb--Luttinger inequality in matrix spaces based on a novel fiber-symmetrization method.

Findings

Established a new inequality for convex bodies in matrix spaces.

Applied the inequality to quasi-concave functions.

Provided potential applications in analysis and geometry.

Abstract

We consider convex bodies in $M_{n,m}(\mathbb R)$, the space of matrices of $n$-rows and $m$-columns. A special case of fiber-symmetrization in $M_{n,m}(\mathbb R)$ was recently introduced in [5,6]. We prove a Rogers--Brascamp--Lieb--Luttinger type inequality with respect to this symmetrization, for quasi-concave functions and provide some applications.