The Steiner Problem for Infinitely Many Points

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.