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Metric Spaces in Which Many Triangles Are Degenerate

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

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 $7n^2/6$ suitably placed degenerate triangles suffice. However, fewer than $n(n-1)/2$ degenerate triangles, no matter how cleverly placed, never suffice.

Metric Spaces in Which Many Triangles Are Degenerate

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 points, fewer than suitably placed degenerate triangles suffice. However, fewer than degenerate triangles, no matter how cleverly placed, never suffice.
Paper Structure (5 theorems, 4 equations)

This paper contains 5 theorems, 4 equations.

Key Result

Theorem 1

If, in a metric space with at least five points, all triangles are degenerate, then this metric space is isometric to a subset of the real line with the usual Euclidean metric.

Theorems & Definitions (7)

  • Theorem 1: Richmond and Richmond
  • Theorem 2
  • Lemma 3
  • Theorem 4
  • proof
  • Theorem 5
  • proof