Basic metric geometry of the bottleneck distance

Abstract

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