On the Structure of Discrete Metric Spaces Isometric to Circles

摘要

A metric space (X, D) is called circular if it is isometric to a subspace of a metric circle, that is, a circle in which distances are measured by the length of the shorter arc connecting them. We show that the following three conditions are equivalent: (1) X is circular, (2) every 4-point subset of X can be labeled as a, b, c, d so that D(a, b) + D(b, c) = D(a,c), D(b,c) + D(c,d) = D(b,d) holds, and (3) every 4-point subset of X is circular.