Polytopes and the mean value property

LetP be any (not necessarily convex nor connected) solid polytope in then-dimensional Euclidean space ℝ_n, and letP(k) be thek-skeleton of P. LetH _P(k) be the set of all continuous functions satisfying the mean value property with respect toP (k). For anyk = 0,1,...,n, we show thatH _P(k) is a finite-dimensional linear space of polynomials. This settles an open problem posed by Friedman and Littman [37] in 1962. Moreover, we show that ifP admits ample symmetry, thenH _P(k) is a finite-dimensional linear space of harmonic polynomials. Some interesting examples are also given