Metric Spaces in Which Many Triangles Are Degenerate

2024/01/10 by Vašek Chvátal, Noé de Rancourt, Chvátal, Vašek +7
Mathematics · #30L99 #51F99 #54E99 #Advanced Banach Space Theory #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2401.05259

openalex publication_date 2024/01/10 · openalex created_date 2024/01/13 · openalex updated_date 2026/07/28

Abstract

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 7n²/6 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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