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,..., xn 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) = supμ ∫X ∫Xd2(x, y)dμ(x)dμ(y), where the supremum is taken over all regular Borel probability measures μ on X.
Citation
Reinhard Wolf. "On the average distance property and certain energy integrals." Ark. Mat. 35 (2) 387 - 400, October 1997. https://doi.org/10.1007/BF02559976
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