A sharp multidimensional Hermite-Hadamard inequality

Let $\Omega \subset \mathbb{R}^d$, $d \geq 2$, be a bounded convex domain and $f\colon \Omega \to \mathbb{R}$ be a non-negative subharmonic function. In this paper we prove the inequality \[ \frac{1}{|\Omega|}\int_\Omega f(x)\,dx \leq \frac{d}{|\partial\Omega|}\int_{\partial\Omega} f(x)\,d\sigma(x)\,. \] Equivalently, the result can be stated as a sharp bound for gradient of the Saint Venant torsion function. Specifically, if $\Omega \subset \mathbb{R}^d$ is a bounded convex domain and $u$ is the solution of $-\Delta u =1$ with homogeneous Dirichlet boundary conditions, then \[ \|\nabla u\|_{L^\infty(\Omega)} < d\frac{|\Omega|}{|\partial\Omega|}\,. \] Moreover, both inequalities are sharp in the sense that if the constant $d$ is replaced by something smaller there exist convex domains for which the inequalities fail. This improves upon the recent result that the optimal constant is bounded from above by $d^{3/2}$ due to Beck et al. A crucial ingredient in our proof of the sharp bounds is the Brascamp-Lieb-Luttinger inequality for convex domains with fixed inradius proved by M\'endez-Hern\'andez.