Improved bounds for Hermite-Hadamard inequalities in higher dimensions

Let $Omega subset mathbb{R}^n$ be a convex domain and let $f:Omega ightarrow mathbb{R}$ be a positive, subharmonic function (i.e. $Delta f geq 0$). Then $$ rac{1}{|Omega|} int_{Omega}{f dx} leq rac{c_n}{ |partial Omega| } int_{partial Omega}{ f dsigma},$$ 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$: $$ rac{|partial Omega_1|}{|Omega_1|} rac{| Omega_2|}{|partial Omega_2|} leq n.$$