Metric Spaces in Which Many Triangles Are Degenerate
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910293495382016 |
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| author | Chvátal, Vašek de Rancourt, Noé Quintero, Guillermo Gamboa Kantor, Ida Szabó, Péter G. N. |
| author_facet | Chvátal, Vašek de Rancourt, Noé Quintero, Guillermo Gamboa Kantor, Ida Szabó, Péter G. N. |
| contents | Richmond and Richmond (American Mathematical Monthly 104 (1997), 713--719) proved the following theorem: If, in a metric space with at least five points, all triangles are degenerate, then the space is isometric to a subset of the real line. We prove that the hypothesis is unnecessarily strong: In a metric space on $n$ points, fewer than $7n^2/6$ suitably placed degenerate triangles suffice. However, fewer than $n(n-1)/2$ degenerate triangles, no matter how cleverly placed, never suffice. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_05259 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Metric Spaces in Which Many Triangles Are Degenerate Chvátal, Vašek de Rancourt, Noé Quintero, Guillermo Gamboa Kantor, Ida Szabó, Péter G. N. Combinatorics Metric Geometry 30L99, 51F99, 54E99 Richmond and Richmond (American Mathematical Monthly 104 (1997), 713--719) proved the following theorem: If, in a metric space with at least five points, all triangles are degenerate, then the space is isometric to a subset of the real line. We prove that the hypothesis is unnecessarily strong: In a metric space on $n$ points, fewer than $7n^2/6$ suitably placed degenerate triangles suffice. However, fewer than $n(n-1)/2$ degenerate triangles, no matter how cleverly placed, never suffice. |
| title | Metric Spaces in Which Many Triangles Are Degenerate |
| topic | Combinatorics Metric Geometry 30L99, 51F99, 54E99 |
| url | https://arxiv.org/abs/2401.05259 |