Arsen Galstyan | Searcharxiv

arXiv2026

Minimal Parametric Networks in Hyperspaces and their Properties

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.

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arXiv2025

Curves in hyperspaces obtained by intersection of $r$-neighborhoods with a fixed subset

The present paper generalizes the result from one of the papers by Galstyan. Namely, we consider two nonempty subsets $A$ and $B$ of a metric space $X$, and construct one-parametric family $F_r$ of subsets obtained by intersection between $B$ and closed $r$-neighborhood of $A$, where $r$ is bigger than the infimum distance between the sets $A$ and $B$. In the case where $B$ is compact, we show that this intersection, considered as a mapping, is right semicontinuously on $r$ in the topology generated by Hausdorff distance. Moreover, if $A$ and $B$ are convex subsets of a normed space $X$, then we prove that $F_r$ depends continuously on $r$ in such topology if and only if the Hausdorff distance between different sets $F_r$ is finite. We also show that for normed spaces $X$ of dimension $2$ or less, the latter condition is automatically fulfilled. For dimension $3$ and hence for bigger ones, we construct an example in which the Hausdorff distance between different $F_r$ is always infinite.

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