The Isometry Theorem

Abstract

This chapter contains a proof of the Isometry Theorem, which asserts that the map from persistence module to persistence diagram is an isometry with respect to the interleaving metric (on modules) and the bottleneck metric (on diagrams). The theorem is valid for the class of q-tame persistence modules. The theorem falls naturally into two parts: the converse stability theorem of Lesnick (the map does not decrease distances), and the stability theorem of Cohen-Steiner, Edelsbrunner and Harer (the map does not increase distances). We finish with a stability theorem for diagrams of rectangle measures. This leads to a very general statement of stability for arbitrary persistence modules.