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 $f:\Omega \rightarrow {\mathbb {R}}$ be a convex function satisfying $f \big |_{\partial \Omega } \ge 0$, then

$$\begin{aligned} \frac{1}{|\Omega |} \int _{\Omega }{f ~\text {d} {\mathcal {H}}^n} \le \frac{2 \pi ^{-1/2} n^{n+1}}{|\partial \Omega |} \int _{\partial \Omega }{f~\text {d} {\mathcal {H}}^{n-1}}. \end{aligned}$$

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, $f:\Omega \rightarrow {\mathbb {R}}_{}$ is subharmonic, i.e., $\Delta f \ge 0$, and $f \big |_{\partial \Omega } \ge 0$, then

$$\begin{aligned} \int _{\Omega }{f~ \text {d} {\mathcal {H}}^2} \le c \cdot \text{ inradius }(\Omega ) \int _{\partial \Omega }{ f ~\text {d}{\mathcal {H}}^{1}}. \end{aligned}$$

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 $\delta$.