Calculus of Variations and Geometric Measure Theory

E. Paolini - E. Stepanov

Existence and regularity results for the Steiner problem

created by stepanov on 18 Jul 2009
modified by paolini on 26 May 2023


Published Paper

Inserted: 18 jul 2009
Last Updated: 26 may 2023

Journal: Calc. Var. Partial Diff. Equations
Volume: 46
Number: 3
Pages: 837-860
Year: 2012
Doi: 10.1007/s00526-012-0505-4


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 $\mathcal{H}$) among the class of sets \[ \mbox{St}(C) :=\{S\subset X\, :\, S\cup C \mbox{ is connected}\}. \] In this paper we study topological regularity results for solutions of this problem in such a general setting. We further provide conditions on existence of minimizers and study the relationships of the above setting with the other similar problem formulations. At last, we provide some applications to locally minimal sets.

Keywords: Steiner problem, minimal connection, geodesic problem