Let \(P\) be an \(n\)-dimensional regular simplex in \({\mathbb{R}}^{n}\) centered at the origin. The \(k\)-skeleton of \(P\) for \(k=1,...,n\) is denoted by \(P\left( k\right) \). The author proved in his earlier publication [Discrete Comput. Geom. 17, No. 2, 163-189 (1997; Zbl 0872.39014)] that the set \({\mathcal H}_{P\left( k\right) }\) of continuous functions in \({\mathbb{R}}^{n}\) satisfying the mean value property with respect to \(P\left( k\right) \) is a finite-dimensional linear subspace of harmonic polynomials. In this paper the author determines the function space \({\mathcal H}_{P\left( k\right) }\) explicitly using combinatorial and group theoretic arguments. The cases \(k=0,n-1,n\) have earlier been considered by \textit{L. Flatto} [Am. J. Math. 85, 248-270 (1963; Zbl 0145.37103)].