Metric spaces in which many triangles are degenerate

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.