Let P be any (not necessarily convex nor connected) solid polytope in the n-dimensional Euclidean space ℝn, and let P(k) be the k-skeleton of P. Let ℋP(k) be the set of all continuous functions satisfying the mean value property with respect to P(k). For any k = 0, 1, . . . , n, we show that ℋP(k) is a finite-dimensional linear space of polynomials. This settles an open problem posed by Friedman and Littman in 1962. Moreover, we show that if P admits ample symmetry, then ℋP(k) is a finite-dimensional linear space of harmonic polynomials. Some interesting examples are also given.
