Basic Metric Geometry of the Bottleneck Distance

Abstract

Given a metric pair $(X,A)$, i.e. a metric space $X$ and a distinguished closed set $A\subset X$, one may construct in a functorial way a pointed pseudometric space ${\mathcal{D}}_{\infty }(X,A)$ of persistence diagrams equipped with the bottleneck distance. We investigate the basic metric properties of the spaces ${\mathcal{D}}_{\infty }(X,A)$ and obtain characterizations of their metrizability, completeness, separability, and geodesicity.

Final version. To appear in Proceedings of the AMS

References in corpus (1)

• Virtual persistence diagrams, signed measures, Wasserstein distances, and Banach spaces

Cited by in corpus (1)

• Grounded persistent path homology: a stable, topological descriptor for weighted digraphs