Basic metric geometry of the bottleneck distance

Mauricio Che, Fernando Galaz-García, Luis Guijarro, Ingrid Membrillo Solis, Motiejus Valiunas

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.