Metric spaces in which many triangles are degenerate

2022/09/28 by Vašek Chvátal, Chvátal, Vašek, Ida Kantor +1 · 1 citation
Computer Science · Mathematics · Engineering · #Digital Image Processing Techniques #Mathematics and Applications #Advanced Numerical Analysis Techniques

paper · pdf · doi:10.48550/arxiv.2209.14361

Abstract

Richmond and Richmond (Amer. Math. 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 fact, Θ(n²) suitably placed degenerate triangles suffice.

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