Suppose $\Omega \subset \mathbb{R}^n$ is some bounded, convex set. For which domains $\Omega$ is it true that for every convex function $f:\Omega \rightarrow \mathbb{R}$ the average of the function in the domain is dominated by the average of the function on the boundary $$ \frac{1}{|\Omega|} \int_{\Omega} f(x) dx \leq \frac{1}{|\partial \Omega|} \int_{\partial \Omega} f(x) dx \quad ?$$ This inequality is trivially true on an interval $\Omega = [a,b]$ (this case is sometimes known as the Hermite-Hadamard inequality) and it's not too hard to see that it's true on the unit ball. However, starting in $n \geq 2$ dimensions, there are also domains for which the inequality fails and, indeed, the inequality will `generically' be false.

There is a particularly fun way to think about this that I learned from a paper of Pasteczka. Affine functions $f(x) = \left\langle a,x\right\rangle + b$ where $a \in \mathbb{R}^n, b \in \mathbb{R}$ have the property that both $f$ and $-f$ are convex and therefore the inequality, if true, has to an equation for affine functions. This has an immediate geometric implication.

Lemma (Pasteczka). If the inequality holds, then the center of mass of $\Omega$ and the center of mass of $\partial \Omega$ have to coincide.

Pasteczka then conjectures that the converse might be true.

Question. Does the inequality hold on every convex domain for which the center of mass of $\Omega$ and $\partial \Omega$ coincide?

I was never quite sure what to believe: on the one hand, it does seem like the statement would be a little bit too good to be true. On the other hand, if the centers of mass agree, one can always add any affine function to $f$ without changing the truth of the inequality. In any case, if true, it would certainly be a very nice Theorem, so I figured it might be a good question for mathoverflow.