# Improved Bounds for Hermite-Hadamard Inequalities in Higher Dimensions

## TL;DR

This paper improves bounds on Hermite-Hadamard inequalities for positive, subharmonic functions in higher dimensions, providing sharper constants and a new geometric inequality relating convex domains.

## Contribution

It establishes new upper and lower bounds for the constants in Hermite-Hadamard inequalities in higher dimensions and derives a novel geometric inequality for convex domains.

## Key findings

- Upper bound for constant c_n: c_n ≤ 2n^{3/2}
- Lower bound for c_n: c_n ≥ n-1
- New geometric inequality relating convex domains

## 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. $\Delta f \geq 0$). Then $$ \frac{1}{|\Omega|} \int_{\Omega}{f dx} \leq \frac{c_n}{ |\partial \Omega| } \int_{\partial \Omega}{ f d\sigma},$$ where $c_n \leq 2n^{3/2}$. This inequality was previously only known for convex functions with a much larger constant. We also show that the optimal constant satisfies $c_n \geq n-1$. 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$: $$ \frac{|\partial \Omega_1|}{|\Omega_1|} \frac{| \Omega_2|}{|\partial \Omega_2|} \leq n.$$