Minimal Parametric Networks in Hyperspaces and their Properties

TL;DR

This paper explores minimal parametric networks in hyperspaces of closed sets within metric spaces, focusing on their properties, solution structures, and conditions for Hausdorff distance realization.

Contribution

It introduces a new perspective by linking minimal parametric networks to the Fermat--Steiner problem in hyperspaces and generalizes results on $d$-far points.

Findings

Minimal parametric networks are nontrivial only within finiteness classes.

Interior vertices of networks relate to Fermat--Steiner problem solutions.

Conditions for realizing one-sided Hausdorff distances are established.

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 $d$-far points in the case of convex boundary sets are also generalized. Namely, conditions are shown under which realizing one-sided Hausdorff distances holds.