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](https://arxiv.org/pdf/1804.03688.pdf). 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.