# The Steiner Problem for Infinitely Many Points

### Emanuele Paolini

Università degli Studi di Firenze, Italy### L. Ulivi

Università degli Studi di Firenze, Italy

## 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.

## Cite this article

Emanuele Paolini, L. Ulivi, The Steiner Problem for Infinitely Many Points. Rend. Sem. Mat. Univ. Padova 124 (2010), pp. 43–56

DOI 10.4171/RSMUP/124-3