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 $$\delta $$.