Polytopes and the mean value property (Q677019)

As main result the author offers an answer to an open problem posed by \textit{A. Friedman} and \textit{W. Littman} [Trans. Am. Math. Soc. 102, 167-180 (1962; Zbl 0103.32201)]: Let \(P\) be any finite union of closed convex solid polytopes in the \(n\)-dimensional real space. Then the set of all continuous functions, assuming at each point their mean value with respect to the union of those faces of \(P\) whose dimension is at most \(k\), is a finite dimensional linear space of polynomials (of harmonic polynomials if \(P\) satisfies certain symmetry properties).