On the average distance property and certain energy integrals

Abstract

One of our main results is the following: Let X be a compact connected subset of the Euclidean space Rⁿ and r(X, d₂) the rendezvous number of X, where d₂ denotes the Euclidean distance in Rⁿ. (The rendezvous number r(X, d₂) is the unique positive real number with the property that for each positive integer n and for all (not necessarily distinct)x₁, x₂,..., x_n in X, there exists some x in X such that $(1/n)\sum\nolimits_{i = 1}^n {d_2 (x_i ,x)} = r(X,d_2 )$ .) Then there exists some regular Borel probability measure μ₀ on X such that the value of ∫_Xd₂(x, y)dμ₀ (y) is independent of the choice x in X, if and only if r(X, d₂) = sup_μ ∫_X ∫_Xd₂(x, y)dμ(x)dμ(y), where the supremum is taken over all regular Borel probability measures μ on X.