The Fermat-Steiner problem in the space of compact subsets of endowed with the Hausdorff metric

Abstract

The Fermat-Steiner problem consists in finding all points in a metric space at which the sum of the distances to fixed points of attains its minimum value. This problem is studied in the metric space of all nonempty compact subsets of the Euclidean space , and the are pairwise disjoint finite sets in . The set of solutions of this problem (which are called Steiner compact sets) falls into different classes in accordance with the distances to the . Each class contains an inclusion-greatest element and inclusion-minimal elements (a maximal Steiner compact set and minimal Steiner compact sets, respectively). We find a necessary and sufficient condition for a compact set to be a minimal Steiner compact set in a given class, provide an algorithm for constructing such compact sets and find a sharp estimate for their cardinalities. We also put forward a number of geometric properties of minimal and maximal compact sets. The results obtained can significantly facilitate the solution of specific problems, which is demonstrated by the well-known example of a symmetric set , for which all Steiner compact sets are asymmetric. The analysis of this case is significantly simplified due to the technique developed. Bibliography 16 titles.