Strengthened volume inequalities for L_p zonoids of even isotropic measures

TL;DR

This paper improves volume inequalities for L_p zonoids of even isotropic measures and their duals, introduces a stronger Brascamp-Lieb inequality, and provides a stability version of the reverse isoperimetric inequality.

Contribution

It presents strengthened inequalities for L_p zonoids, a novel version of the Brascamp-Lieb inequality, and a stability result for the reverse isoperimetric inequality.

Findings

Enhanced volume inequalities for L_p zonoids and their duals.

A stronger Brascamp-Lieb inequality for approximating Gaussians.

A stability version of the reverse isoperimetric inequality.

Abstract

We strengthen the volume inequalities for L_p zonoids of even isotropic measures and for their duals, which are due to Ball, Barthe and Lutwak, Yang, Zhang. Along the way, we prove a stronger version of the Brascamp-Lieb inequality for a family of functions that can approximate arbitrary well some Gaussians when equality holds. The special case p=\infty yields a stability version of the reverse isoperimetric inequality for centrally symmetric bodies.