The Hermite-Hadamard inequality in higher dimensions

Abstract

The Hermite-Hadamard inequality states that the average value of a convex function on an interval is bounded from above by the average value of the function at the endpoints of the interval. We provide a generalization to higher dimensions: let $\Omega \subset {\mathbb{R}}^n$ be a convex domain and let $f:\Omega \to \mathbb{R}$ be a convex function satisfying $f{|}_{\partial \Omega }\ge 0$, then $$\frac{1}{|\Omega |}{\int }_{\Omega }fd{\mathcal{H}}^n\le \frac{2{\pi }^{-1/2}{n}^{n+1}}{|\partial \Omega |}{\int }_{\partial \Omega }fd{\mathcal{H}}^{n-1}.$$ The constant $2{\pi }^{-1/2}{n}^{n+1}$ is presumably far from optimal, however, it cannot be replaced by 1 in general. We prove slightly stronger estimates for the constant in two dimensions where we show that $9/8\le c_2\le 8$. We also show, for some universal constant $c>0$, if $\Omega \subset {\mathbb{R}}^2$ is simply connected with smooth boundary, $f:\Omega \to {\mathbb{R}}_{}$ is subharmonic, i.e. $\Delta f\ge 0$, and $f{|}_{\partial \Omega }\ge 0$, then $$ \int_Ω{f~ d \mathcal{H}^2} \leq c \cdot \mbox{inradius}(Ω) \int_{\partial Ω}{ f ~d\mathcal{H}^{1}}.$$ We also prove that every domain $\Omega \subset {\mathbb{R}}^n$ whose boundary is 'flat' at a certain scale $\delta$ admits a Hermite-Hadamard inequality for all subharmonic functions with a constant depending only on the dimension, the measure $|\Omega |$ and the scale $\delta$.

References in corpus (1)

• Jensen-type geometric shapes