Regular Polytopes and Harmonic Polynomials

Abstract

In this paper we study the following problem originally proposed by Walsh (8). To determine the class of functions f(x) continuous in a given n-dimensional region R and having the property that the value of f(x) be equal to the average of f(x) over the vertices of all sufficiently small regular polytopes similar to a given one, which are centred at x. This problem has been studied by several mathematicians (1; 6; 8) and has been completely solved except for the four-dimensional regular polytopes {3, 4, 3}, {3, 3, 5}, {5, 3, 3} (see 3, p. 129, for the meaning of these symbols) and the n-dimensional cube. In each case, the class of functions is identical with a class of harmonic polynomials which can be specified. In § 2, we solve the problem for the four-dimensional figures, thus leaving the problem open only for the n-dimensional cube.