Geodesic
Regular Polytopes and Harmonic Polynomials
Canadian journal of mathematics, Tome 22 (1970) no. 1, pp. 7-21

Voir la notice de l'article provenant de la source Cambridge University Press

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. 
Flatto, Leopold; Wiener, Sister Margaret M. Regular Polytopes and Harmonic Polynomials. Canadian journal of mathematics, Tome 22 (1970) no. 1, pp. 7-21. doi: 10.4153/CJM-1970-002-2
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