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