A dimension-free Hermite–Hadamard inequality via gradient estimates for the torsion function

Abstract

Let Ω ⊂ R n \Omega \subset \mathbb {R}^n be a convex domain, and let f : Ω → R f:\Omega \rightarrow \mathbb {R} be a subharmonic function, Δ f ≥ 0 \Delta f \geq 0 , which satisfies f ≥ 0 f \geq 0 on the boundary ∂ Ω \partial \Omega . Then ∫ Ω f d x ≤ | Ω | 1 n ∫ ∂ Ω f d σ . \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, Δ u = − 1 \Delta u = -1 with Dirichlet boundary conditions, which is of independent interest.