Curves in hyperspaces obtained by intersection of $r$-neighborhoods with a fixed subset
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866908696663031808 |
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| author | Galstyan, Arsen Tuzhilin, Alexey |
| author_facet | Galstyan, Arsen Tuzhilin, Alexey |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_06327 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Curves in hyperspaces obtained by intersection of $r$-neighborhoods with a fixed subset Galstyan, Arsen Tuzhilin, Alexey Metric Geometry 46B20, 51F99, 52A07, 52A40, 52A41, 46B50 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. |
| title | Curves in hyperspaces obtained by intersection of $r$-neighborhoods with a fixed subset |
| topic | Metric Geometry 46B20, 51F99, 52A07, 52A40, 52A41, 46B50 |
| url | https://arxiv.org/abs/2512.06327 |