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.
The Steiner problem for infinitely many points / E. Paolini; L. Ulivi. - In: RENDICONTI DEL SEMINARIO MATEMATICO DELL'UNIVERSITA' DI PADOVA. - ISSN 0041-8994. - STAMPA. - 124:(2010), pp. 43-56. [10.4171/RSMUP/124-3]
The Steiner problem for infinitely many points
PAOLINI, EMANUELE;
2010
Abstract
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.File | Dimensione | Formato | |
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