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

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a convex domain and let $f:\Omega \rightarrow $\mathbb{R}$$beasubharmonicfunction,$\Delta f \geq 0$, which satisfies $f\ge 0$ on the boundary $\mathrm{\partial }\mathrm{\Omega }$. Then $${\int }_{\mathrm{\Omega }}fdx\le \mathrm{\mid }\mathrm{\Omega }{\mathrm{\mid }}^{\frac{1}{n}}{\int }_{\mathrm{\partial }\mathrm{\Omega }}fd\sigma \mathrm{.}$$ 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.