Universality of the Bottleneck Distance for Extended Persistence Diagrams

The extended persistence diagram is an invariant of piecewise linear functions, introduced by Cohen-Steiner, Edelsbrunner, and Harer. The bottleneck distance has been introduced by the same authors as an extended pseudometric on the set of extended persistence diagrams, which is stable under perturbations of the function. We address the question whether the bottleneck distance is the largest possible stable distance, providing an affirmative answer.Comment: 20 pages + 12 pages appendix, 18 figures, LaTeX; removal of appendix on "stable functors on M" which has moved to arXiv:2108.09298, added and improved figures, added a note of caution regarding variants of the bottleneck distance, rewrote the proof of lemma 4.6 (formerly lemma 4.4), added appendix B including the connection to the original definition of extended persistence, several minor edit