Isobarycentric Inequalities

TL;DR

This paper proves a new isoperimetric inequality involving barycenters and measures, advancing the understanding of geometric measure problems and their connections to log-concave random variables.

Contribution

It introduces a novel inequality relating measure, barycenter, and halfspaces, and links it to the Log-Minkowski inequality for specific convex bodies.

Findings

Halfspaces maximize measure for fixed barycenter.

Progress on Henk and Pollehn's problem related to Log-Minkowski inequality.

New inequalities and conjectures for log-concave random variables.

Abstract

We prove the following isoperimetric type inequality: Given a finite absolutely continuous Borel measure on ${\mathbb R}^n$, halfspaces have maximal measure among all subsets with prescribed barycenter. As a consequence, we make progress towards a solution to a problem of Henk and Pollehn, which is equivalent to a Log-Minkowski inequality for a parallelotope and a centered convex body. Our probabilistic approach to the problem also gives rise to several inequalities and conjectures concerning the truncated mean of certain log-concave random variables.