Hermite--Hadamard inequalities for nearly-spherical domains

A conjecture of Pasteczka, generalizing the classical Hermite–Hadamard Inequality, states that if $\mathrm{\Omega }\subseteq {\mathbb{R}}^d$ is a compact convex domain such that $\mathrm{\Omega }$ and $\mathrm{\partial }\mathrm{\Omega }$ have the same center of mass, then for every convex function $f:\mathrm{\Omega }\to {\mathbb{R}}^d$, the average value of $f$ on $\mathrm{\Omega }$ is less than or equal to the average value of $f$ on $\mathrm{\partial }\mathrm{\Omega }$. Pasteczka proved this conjecture for the case where $\mathrm{\Omega }$ 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 $(d+1)\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }\mathrm{/}\mathrm{\mid }\mathrm{\partial }\mathrm{\Omega }\mathrm{\mid }$ away from all hyperplanes tangent to $\mathrm{\partial }\mathrm{\Omega }$.