Metric spaces in which many triangles are degenerate

TL;DR

This paper shows that in metric spaces, having a quadratic number of carefully arranged degenerate triangles is enough to ensure the space is essentially a subset of the real line, relaxing previous conditions.

Contribution

The authors demonstrate that fewer degenerate triangles than previously thought are sufficient to characterize metric spaces as subsets of the real line.

Findings

Quadratic number of degenerate triangles suffices

Relaxation of previous strong hypotheses

Characterization of metric spaces as real line subsets

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, $\Theta(n^2)$ suitably placed degenerate triangles suffice.