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 functions. We address the question whether the bottleneck distance is the largest possible stable distance, providing an affirmative answer. Finally, we contrast the bottleneck distance with the interleaving distance of sheaves by showing that the interleaving distance of sheaves is not intrinsic, let alone universal.