Set-valued metrics and generalized Hausdorff distances
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915852559843328 |
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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 $X$. We show that Hausdorff distance $d_H$ belongs to a specific family of real-valued distances on $BCl(X)$, each of which can be expressed as the composition $μ\circ d_{sv}$ of a topology inducing set-valued function $d_{sv}:BCl(X)^2\rightarrow \mathcal{P}(Z)$ and a real-valued set-function $μ:Σ\subset\mathcal{P}(Z)\rightarrow\mathbb{R}$. With this observation, we construct several associated classes of inter-set distances, called set-valued metrics and generalized Hausdorff distances. Our constructions are both explicit and adaptable, and the resulting distance classes are expected to cover most practical applications involving distance between sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_10815 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Set-valued metrics and generalized Hausdorff distances Akofor, Earnest General Topology Primary 54E05, Secondary 54B20 54C60 Let $X$ be a metric space and $BCl(X)$ the collection of nonempty bounded closed subsets of $X$. We show that Hausdorff distance $d_H$ belongs to a specific family of real-valued distances on $BCl(X)$, each of which can be expressed as the composition $μ\circ d_{sv}$ of a topology inducing set-valued function $d_{sv}:BCl(X)^2\rightarrow \mathcal{P}(Z)$ and a real-valued set-function $μ:Σ\subset\mathcal{P}(Z)\rightarrow\mathbb{R}$. With this observation, we construct several associated classes of inter-set distances, called set-valued metrics and generalized Hausdorff distances. Our constructions are both explicit and adaptable, and the resulting distance classes are expected to cover most practical applications involving distance between sets. |
| title | Set-valued metrics and generalized Hausdorff distances |
| topic | General Topology Primary 54E05, Secondary 54B20 54C60 |
| url | https://arxiv.org/abs/2503.10815 |