# The Steiner problem for infinitely many points

created by paolini on 19 Aug 2009
modified on 30 Nov 2016

[BibTeX]

Published Paper

Inserted: 19 aug 2009
Last Updated: 30 nov 2016

Journal: Rend. Sem. Mat. Univ. Padova
Volume: 124
Pages: 43-56
Year: 2010
Doi: 10.4171/RSMUP/124-3

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 recovery 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.

Keywords: steiner, minimal