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

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.