Inequalities for convex functions of random points inside and on the boundary of convex bodies

Abstract

Let $K$ be a convex body in ${\mathbb{R}}^d$, and let $I$ and $B$ be random points uniformly distributed inside $K$ and on its boundary, respectively. We prove that if $d=2$ and $\mathbb{E}I=\mathbb{E}B$, or if $K$ is a circumscribed polytope with the center of the inscribed sphere coinciding with $\mathbb{E}I=\mathbb{E}B$, then $I$ is dominated by $B$ in the convex order. As a consequence, for any function $\phi$ convex in each argument, the expectation $\mathbb{E}\phi (I_1,\dots ,I_k)$ does not exceed $\mathbb{E}\phi (B_1,\dots ,B_k)$. This yields, in particular, an inequality between the moments of random chords $\mathbb{E}|I_1-I_2{|}^p⩽\mathbb{E}|B_1-B_2{|}^p$ for all $p⩾1$, confirming the Zaporozhets--Tarasov conjecture for the indicated class of bodies, and extends to inequalities for mean volumes of random simplices.