Existence and regularity results for the Steiner problem

Given a complete metric space X and a compact set ${C\subset X}$ , the famous Steiner (or minimal connection) problem is that of finding a set S of minimum length (one-dimensional Hausdorff measure ${\mathcal H^1)}$ ) among the class of sets $$\mathcal{S}t(C) \,:=\{S\subset X\colon S \cup C \,{\rm is connected}\}.$$ In this paper we provide conditions on existence of minimizers and study topological regularity results for solutions of this problem. We also study the relationships between several similar variants of the Steiner problem. At last, we provide some applications to locally minimal sets.