  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$:
  $$ \frac{|\partial \Omega_1|}{|\Omega_1|} \frac{| \Omega_2|}{|\partial
\Omega_2|} \leq n.$$
