On the average distance property and certain energy integrals

Reinhard Wolf Institut für Mathematik, Universität Salzburg

TBD mathscidoc:1701.332880

Arkiv for Matematik, 35, (2), 387-400, 1996.8
One of our main results is the following: Let$X$be a compact connected subset of the Euclidean space$R$^{$n$}and$r(X, d$_{2}) the rendezvous number of$X$, where$d$_{2}denotes the Euclidean distance in$R$^{$n$}. (The rendezvous number$r(X, d$_{2}) is the unique positive real number with the property that for each positive integer$n$and for all (not necessarily distinct)$x$_{1},$x$_{2},...,$x$_{$n$}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 μ_{0}on$X$such that the value of ∫_{$X$}$d$_{2}($x, y$)$d$μ_{0}($y$) is independent of the choice$x$in$X$, if and only if$r$($X, d$_{2}) = sup_{μ}∫_{$X$}∫_{$X$}$d$_{2}($x, y$)$d$μ($x$)$d$μ($y$), where the supremum is taken over all regular Borel probability measures μ on$X$.
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  title={On the average distance property and certain energy integrals},
  author={Reinhard Wolf},
  booktitle={Arkiv for Matematik},
Reinhard Wolf. On the average distance property and certain energy integrals. 1996. Vol. 35. In Arkiv for Matematik. pp.387-400. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203551574281689.
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