On Quasiconvexity of Precompact-Subset Spaces
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866916372141834240 |
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| author | Akofor, Earnest |
| author_facet | Akofor, Earnest |
| contents | Let $X$ be a metric space and $BCl(X)$ the collection of nonempty bounded closed subsets of $X$ as a metric space with respect to Hausdorff distance. We study both characterization and representation of Lipschitz paths in $BCl(X)$ in terms of Lipschitz paths in $X$ and in the completion of $X$. We show that a full characterization and representation is possible in any subspace $\mathcal{J}\subset BCl(X)$ that (i) consists of precompact subsets of $X$, (ii) contains the singletons $\{x\}$ for every $x\in X$, and (iii) satisfies $BCl(C)\subset\mathcal{J}$ for every $C\in\mathcal{J}$. When $X$ is geodesic, we investigate quasiconvexity of $\mathcal{J}$ for some instances of $\mathcal{J}$, especially when $\mathcal{J}$ consists of finite subsets of $X$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_07725 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Quasiconvexity of Precompact-Subset Spaces Akofor, Earnest General Topology Primary 52A01, Secondary 54B20 54E05 40A30 03E75 Let $X$ be a metric space and $BCl(X)$ the collection of nonempty bounded closed subsets of $X$ as a metric space with respect to Hausdorff distance. We study both characterization and representation of Lipschitz paths in $BCl(X)$ in terms of Lipschitz paths in $X$ and in the completion of $X$. We show that a full characterization and representation is possible in any subspace $\mathcal{J}\subset BCl(X)$ that (i) consists of precompact subsets of $X$, (ii) contains the singletons $\{x\}$ for every $x\in X$, and (iii) satisfies $BCl(C)\subset\mathcal{J}$ for every $C\in\mathcal{J}$. When $X$ is geodesic, we investigate quasiconvexity of $\mathcal{J}$ for some instances of $\mathcal{J}$, especially when $\mathcal{J}$ consists of finite subsets of $X$. |
| title | On Quasiconvexity of Precompact-Subset Spaces |
| topic | General Topology Primary 52A01, Secondary 54B20 54E05 40A30 03E75 |
| url | https://arxiv.org/abs/2308.07725 |