Mean distances on a sphere (Q550544)

A set of \(m\) point is considered to be scattered on a sphere if each such point \(P\) has a surrounding ``neighborhood'' -- the open set of points closer to \(P\) than to any of the other \(m-1\) points. The mean distance from \(P\) to its neighborhood is defined in the usual way by integration and the mean distance from the whole sphere to the nearest of the \(m\) points is the the weighted mean of the \(m\) neighborhood means. In this paper, the author estimates such mean distances for various choices of \(m\) points scattered on a sphere, including vertex sets of the five regular polyhedra, and these appear to be new. For example, the mean distance of a given regular tetrahedron inscribed in the unit sphere to the nearest vertex is approximately 0.690737280538164. In the final section, the author considers distance as a probability random variable, of which ``expected value'' is a synonym for mean distance.