Rendiconti del Seminario Matematico della Università di Padova


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Volume 124, 2010, pp. 43–56
DOI: 10.4171/RSMUP/124-3

Published online: 2010-12-31

The Steiner Problem for Infinitely Many Points

Emanuele Paolini[1] and L. Ulivi[2]

(1) Università degli Studi di Firenze, Italy
(2) Università degli Studi di Firenze, Italy

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 containing A. 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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Paolini Emanuele, Ulivi L.: The Steiner Problem for Infinitely Many Points. Rend. Sem. Mat. Univ. Padova 124 (2010), 43-56. doi: 10.4171/RSMUP/124-3