Isobarycentric Inequalities

2022/02/15 by Gilboa, Shoni, Haim-Kislev, Pazit, Slomka, Boaz
#FOS: Mathematics #Metric Geometry (math.MG) #Probability (math.PR)

paper · doi:10.48550/arxiv.2202.07527

Abstract

We prove the following isoperimetric type inequality: Given a finite absolutely continuous Borel measure on \mathbb Rⁿ, 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.

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