On the center of distances

Abstract

We introduce the notion of a center of distances of a metric space and use it in a generalization of the theorem by John von Neumann on permutations of two sequences with the same set of cluster points in a compact metric space. This notion is also used to study sets of subsums of some sequences of positive reals, as well for some impossibility proofs. We compute the center of distances of the Cantorval, which is the set of subsums of the sequence 34 ,12 ,316 ,18 ,…,34 n ,24 n ,… 34,12,316,18,…,34n,24n,… , and for other related subsets of the reals.