Minimal Parametric Networks in Hyperspaces and their Properties
arXiv:2604.28004
The paper studies minimal parametric networks in hyperspaces of closed subsets of metric spaces equipped with the Hausdorff distance, linking their interior vertices to solutions of the Fermat–Steiner problem and describing solution classes and conditions for one‑sided Hausdorff distances.
Abstract
This work investigates minimal parametric networks in hyperspaces of closed subsets of metric spaces endowed with the Hausdorff distance. It is shown that the problems of finding such networks are nontrivial only within finiteness classes, where all Hausdorff distances between elements are finite. It is demonstrated that when studying the properties of minimal parametric networks, it is convenient to view their interior vertices as solutions of the Fermat--Steiner problem on the adjacent vertices. In this connection, already within the framework of the Fermat--Steiner problem, the structure of solution classes in hyperspaces of closed subsets of metric spaces is described. Results on the existence of -far points in the case of convex boundary sets are also generalized. Namely, conditions are shown under which realizing one-sided Hausdorff distances holds.