On the Approximation of Certain Mass Distributions Appearing in Distance Geometry

Abstract

Let X be a compact subset of the n-dimensional Euclidean space R n . A theorem of G. Bjorck implies the existence of a unique probability measure μ0 which maximizes the value ∫ X ∫ X d2(x, y) dμ(x) dμ(y), where μ ranges over all probability measures on X and d2 denotes the Euclidean distance on R n . 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.