arXiv Open Access 2024

Metric Spaces in Which Many Triangles Are Degenerate

Vašek Chvátal Noé de Rancourt Guillermo Gamboa Quintero Ida Kantor Péter G. N. Szabó

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Abstrak

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.

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math.CO math.MG

Penulis (5)

Vašek Chvátal

Noé de Rancourt

Guillermo Gamboa Quintero

Ida Kantor

Péter G. N. Szabó

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Chvátal, V., Rancourt, N.d., Quintero, G.G., Kantor, I., Szabó, P.G.N. (2024). Metric Spaces in Which Many Triangles Are Degenerate. https://arxiv.org/abs/2401.05259

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Tahun Terbit: 2024
Bahasa: en
Sumber Database: arXiv
Akses: Open Access ✓