Hermite--Hadamard inequalities for nearly-spherical domains

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