A symmetry problem

Abstract

The following result is proved: {\bf Theorem.} Let $D\subset {\mathbb{R}}^3$ be a bounded domain homeomorphic to a ball, $|D|$ be its volume, $|S|$ be the surface area of its smooth boundary $S$, $D\subset B_R:=\{x:|x|\le R\}$, and $H_R$ is the set of all harmonic in $B_R$ functions. If $$\frac{1}{|D|}{\int }_{D}hdx=\frac{1}{|S|}{\int }_{S}hds\forall h\in H_R,$$ then $D$ is a ball.