Improved Bounds for Hermite-Hadamard Inequalities in Higher Dimensions

Abstract

Let $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$ be a convex domain and let $f:\hat{I}\text{c◯}\to \mathbb{R}$ be a positive, subharmonic function (i.e. $\hat{I}”f\ge 0$). Then $$\frac{1}{|\hat{I}\text{c◯}|}{\int }_{\hat{I}}\text{c◯}fdx\le \frac{c_n}{|\partial \hat{I}\text{c◯}|}{\int }_{\partial \hat{I}\text{c◯}}fd\ddot{I}ƒ,$$ 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 $\hat{I}{\text{c◯}}_2\subset \hat{I}{\text{c◯}}_1\subset {\mathbb{R}}^n$: $$\frac{|\partial \hat{I}{\text{c◯}}_1|}{|\hat{I}{\text{c◯}}_1|}\frac{|\hat{I}{\text{c◯}}_2|}{|\partial \hat{I}{\text{c◯}}_2|}\le n.$$