Consider a compact, convex domain $\Omega \subset \mathbb{R}^3$ with $|\Omega|=1$ with smooth boundary $\partial \Omega$.

Now consider the electric potential generated by this uniform mass distribution: $\phi = \int_{\Omega} \frac{1}{|x-y|} dx$.
**Question:** I would like to know if there is a relationship between mean curvature $H$ and $\phi$. What I have in mind is an inequality of the form
$\|\phi - \bar \phi\|_{L^p(\partial \Omega)} \leq C \|H - \bar H\|_{L^p(\partial \Omega)}$ where $\bar \phi$ and $\bar H$ denote the averages over $\partial \Omega$ of $\phi$ and $H$ respectively and $p$ can be anything for the time being.

Although one term is much more 'non-local' than the other, for a convex body it seems reasonable that the closer to being an surface of constant mean curvature, the closer one is to being an equipotential surface. This is related to a previous question of mine which ended up going unanswered regarding whether the only convex, compact equipotential surface in $\mathbb{R}^3$ was a sphere or not.