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Grünbaum's inequality for probability measures with concavity properties

Grünbaum's inequality for probability measures with concavity properties

来源:Arxiv_logoArxiv
英文摘要

A celebrated result in convex geometry is Grünbaum's inequality, which quantifies how much volume of a convex body can be cut off by a hyperplane passing through its barycenter. In this work, we establish a series of sharp Grünbaum-type inequalities - with equality characterizations - for probability measures under certain concavity assumptions. Of particular interest is the case of $s$-concave Radon measures. For such measures, we provide a simpler proof of known results and, more importantly, obtain the previously missing equality characterization. This is achieved by gaining new insight into the equality case of their Brunn-Minkowski-type inequality. Moreover, we show that these ``$s$-Grünbaum'' inequalities can hold only when $s > -1$. However, for convex measures on the real line, we prove Grünbaum-type inequalities involving their cumulative distribution function. Finally, by applying the renowned Ehrhard inequality, we deduce an ``Ehrhard-Grünbaum'' inequality for the Gaussian measure on $\mathbb{R}^n$, which improves upon the bound derived from its log-concavity.

Matthieu Fradelizi、Dylan Langharst、Jiaqian Liu、Francisco Marín Sola、Shengyu Tang

数学

Matthieu Fradelizi,Dylan Langharst,Jiaqian Liu,Francisco Marín Sola,Shengyu Tang.Grünbaum's inequality for probability measures with concavity properties[EB/OL].(2025-07-10)[2025-07-23].https://arxiv.org/abs/2507.06759.点此复制

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