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 $f:\Omega \rightarrow \mathbb {R}$ be a subharmonic function, $\Delta f \geq 0$, which satisfies $f \geq 0$ on the boundary $\partial \Omega$. Then \begin{equation*} \int _{\Omega }{f ~dx} \leq |\Omega |^{\frac {1}{n}} \int _{\partial \Omega }{f ~d\sigma }. \end{equation*} 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.