Center and radius of a subset of metric space
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910571699372032 |
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| author | Badra, Akhilesh Singh, Hemant Kumar |
| author_facet | Badra, Akhilesh Singh, Hemant Kumar |
| contents | In this paper, we introduce a notion of the center and radius of a subset A of metric space X. In the Euclidean spaces, this notion can be seen as the extension of the center and radius of open/closed balls. The center and radius of a finite product of subsets of metric spaces, and a finite union of subsets of a metric space are also determined. For any subset A of metric space X, there is a natural question to identify the open balls of X with the largest radius that are entirely contained in A. To answer this question, we introduce a notion of quasi-center and quasi-radius of a subset A of metric space X. We prove that the center of the largest open balls contained in A belongs to the quasi-center of A, and its radius is equal to the quasi-radius of A. In particular, for the Euclidean spaces, we see that the center of largest open balls contained in A belongs to the center of A, and its radius is equal to the radius of A. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_15772 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Center and radius of a subset of metric space Badra, Akhilesh Singh, Hemant Kumar General Topology Primary 54E35, Secondary 54E99 In this paper, we introduce a notion of the center and radius of a subset A of metric space X. In the Euclidean spaces, this notion can be seen as the extension of the center and radius of open/closed balls. The center and radius of a finite product of subsets of metric spaces, and a finite union of subsets of a metric space are also determined. For any subset A of metric space X, there is a natural question to identify the open balls of X with the largest radius that are entirely contained in A. To answer this question, we introduce a notion of quasi-center and quasi-radius of a subset A of metric space X. We prove that the center of the largest open balls contained in A belongs to the quasi-center of A, and its radius is equal to the quasi-radius of A. In particular, for the Euclidean spaces, we see that the center of largest open balls contained in A belongs to the center of A, and its radius is equal to the radius of A. |
| title | Center and radius of a subset of metric space |
| topic | General Topology Primary 54E35, Secondary 54E99 |
| url | https://arxiv.org/abs/2406.15772 |