Gr\"unbaum's inequality for Gaussian and convex probability measures

Abstract

A celebrated result in convex geometry is Gr\"unbaum'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 Gr\"unbaum-type inequalities - with equality characterizations - for probability measures under certain concavity assumptions. As an application, we apply the renowned Ehrhard inequality and deduce an ``Ehrhard-Gr\"unbaum'' inequality for the Gaussian measure on $\mathbb{R}^n$, which improves upon the bound derived from its log-concavity. For $s$-concave Radon measures, our framework provides a simpler proof of known results and, more importantly, yields 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 ``$s$-Gr\"unbaum'' inequalities can hold only when $s > -1$. However, for convex measures on the real line, we prove Gr\"unbaum-type inequalities involving their cumulative distribution function.