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 ⊂ X, one may construct in a functorial way a pointed pseudometric space D∞(X, A) of persistence diagrams equipped with the bottleneck distance. We investigate the basic metric properties of the spaces D∞(X, A) and obtain characterizations of their metrizability, completeness, separability, and geodesicity.