Hermite--Hadamard inequalities for nearly-spherical domains

Abstract

A conjecture of Pasteczka, generalizing the classical Hermite--Hadamard Inequality, states that if $Ω\subseteq \mathbb{R}^d$ is a compact convex domain such that $Ω$ and $\partial Ω$ have the same center of mass, then for every convex function $f: Ω\to \mathbb{R}^d$, the average value of $f$ on $Ω$ is less than or equal to the average value of $f$ on $\partial Ω$. Pasteczka proved this conjecture for the case where $Ω$ is a polytope with an inscribed ball. We generalize this result by proving Pasteczka's conjecture in the case where some point lies at most $(d+1)|Ω|/|\partial Ω|$ away from all hyperplanes tangent to $\partial Ω$.