On the Approximation of Certain Mass Distributions Appearing in Distance Geometry

Let X be a compact subset of the n-dimensional Euclidean space Rn. A theorem of G. Björck implies the existence of a unique probability measure μ0 which maximizes the value ∫X∫Xd2(x, y) dμ(x) dμ(y), where μ ranges over all probability measures on X and d2 denotes the Euclidean distance on Rn. In this paper we introduce and investigate an algorithm which is easy to describe and which inductively constructs a sequence ω = x1, x2,... in X such that ω is uniformly distributed with respect to μ0. Geometrical and topological interpretations and applications, and concrete numerical examples are given.