Let A be a given compact subset of the euclidean space. We consider the problem of finding a compact connected set S of minimal 1- dimensional Hausdorff measure, among all compact connected sets containingA. We prove that when A is a finite set any minimizer is a finite tree with straight edges, thus recovering the classical Steiner Problem. Analogously, in the case when A is countable, we prove that every minimizer is a (possibly) countable union of straight segments.
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Emanuele Paolini, L. Ulivi, The Steiner Problem for Infinitely Many Points. Rend. Sem. Mat. Univ. Padova 124 (2010), pp. 43–56DOI 10.4171/RSMUP/124-3