Suppose \(\Omega\subseteq \mathbb{R}^d\) is a open. A function \(u\in C^2(\Omega)\) is subharmonic if and only it it suffices the mean value property .
Proof
From the proof of the mean value property we already know, that harmonicity implies the boundary version and the boundary version implies the full ball version. Assume
\[ u(x)=\frac{1}{\lvert B_r(x)\rvert} \int_{B_r(x)} u \]for all suitable \(x\in \Omega\) and \(r>0\).
Assume there is a point \(x\in \Omega\) with \(\Delta u(x)<0\). Then there is a ball \(B_r(x)\) such that \(\Delta u<0\) on this ball. Inserting this into the proof of the mean value problem implies
\[ \frac{1}{\lvert B_r(x)\rvert} \int_{B_r(x)} u < u(x) \]