A Dimension-Free Hermite-Hadamard Inequality via Gradient Estimates for the Torsion Function

Abstract

Let $\Omega \subset {\mathbb{R}}^n$ be a convex domain and let $f:\Omega \to \mathbb{R}$ be a subharmonic function, $\Delta f\ge 0$, which satisfies $f\ge 0$ on the boundary $\partial \Omega$. Then $${\int }_{\Omega }fdx\le |\Omega {|}^{\frac{1}{n}}{\int }_{\partial \Omega }fd\sigma .$$ Our proof is based on a new gradient estimate for the torsion function, $\Delta u=-1$ with Dirichlet boundary conditions, which is of independent interest.