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

TL;DR

This paper studies how the intersection of a fixed subset with the r-neighborhood of another subset in a metric space behaves as r varies, establishing conditions for continuity and semicontinuity in different space dimensions.

Contribution

It generalizes previous results by analyzing the continuity properties of intersection families in metric and normed spaces, including new conditions for finite Hausdorff distances.

Findings

Right semicontinuity of the intersection family in compact sets

Continuity of the intersection family in convex sets when Hausdorff distance is finite

Automatic fulfillment of finiteness condition in spaces of dimension two or less

Abstract

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.