  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
$\partial \Omega$.
