Paper detail

A sharp multidimensional Hermite-Hadamard inequality

Let $Ω\subset \mathbb{R}^d$, $d \geq 2$, be a bounded convex domain and $f\colon Ω\to \mathbb{R}$ be a non-negative subharmonic function. In this paper we prove the inequality \[ \frac{1}{|Ω|}\int_Ωf(x)\,dx \leq \frac{d}{|\partialΩ|}\int_{\partialΩ} f(x)\,dσ(x)\,. \] Equivalently, the result can be stated as a bound for the gradient of the Saint Venant torsion function. Specifically, if $Ω\subset \mathbb{R}^d$ is a bounded convex domain and $u$ is the solution of $-Δu =1$ with homogeneous Dirichlet boundary conditions, then \[ \|\nabla u\|_{L^\infty(Ω)} < d\frac{|Ω|}{|\partialΩ|}\,. \] Moreover, both inequalities are sharp in the sense that if the constant $d$ is replaced by something smaller there exist convex domains for which the inequalities fail. This improves upon the recent result that the optimal constant is bounded from above by $d^{3/2}$ due to Beck et al.