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 $\varphi$ convex in each argument, the expectation $\mathbb E \varphi(I_1,\dots,I_k)$ does not exceed $\mathbb E \varphi(B_1,\dots,B_k)$. This yields, in particular, an inequality between the moments of random chords $\mathbb E |I_1 - I_2|^p \leqslant \mathbb E |B_1 - B_2|^p$ for all $p \geqslant 1$, confirming the Zaporozhets--Tarasov conjecture for the indicated class of bodies, and extends to inequalities for mean volumes of random simplices.