Improved Bounds for Hermite–Hadamard Inequalities in Higher Dimensions

Abstract

Let $\Omega \subset {\mathbb {R}}^n$ be a convex domain and let $f:\Omega \rightarrow {\mathbb {R}}$ be a positive, subharmonic function (i.e., $\Delta f \ge 0$). Then

$$\begin{aligned} \frac{1}{|\Omega |} \int _{\Omega }{f \mathrm{{d}}x} \le \frac{c_n}{ |\partial \Omega | } \int _{\partial \Omega }{ f \mathrm{{d}}\sigma }, \end{aligned}$$

where $c_n \le 2n^{3/2}$. This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies $c_n \ge n-1$. As a byproduct, we establish a sharp geometric inequality for two convex domains where one contains the other $\Omega _2 \subset \Omega _1 \subset {\mathbb {R}}^n$:

$$\begin{aligned} \frac{|\partial \Omega _1|}{|\Omega _1|} \frac{| \Omega _2|}{|\partial \Omega _2|} \le n. \end{aligned}$$