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

Let $\Omega \subset \mathbb{R}^n$ be a convex domain and let $f:\Omega \rightarrow \mathbb{R}$ be a subharmonic function, $\Delta f \geq 0$, which satisfies $f \geq 0$ on the boundary $\partial \Omega$. Then $$ \int_{\Omega}{f ~dx} \leq |\Omega|^{\frac{1}{n}} \int_{\partial \Omega}{f ~d\sigma}.$$ 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.