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