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