Metric pairs and tuples in theory and applications

TL;DR

This work extends the Gromov--Hausdorff distance to metric pairs and tuples, establishing that the space of compact metric pairs $(X,A)$ with the distance $d_{\mathop{\mathrm{GH}}}$ is geodesic and separable, and proving an Arzelà--Ascoli-type compactness for relative maps. It proves a Cassorla-type density result showing that 2D manifolds embedded in $\mathbb{R}^3$ with boundary submanifolds densely approximate all compact metric pairs, enriching the link between abstract metric geometry and concrete surface models. The paper also demonstrates the natural appearance and utility of these distances in applications such as hypernetworks, simplicial Hausdorff distance, and persistence-m matching diagrams, and provides surrogate distances and stability results for these settings. Overall, it furnishes a rigorous, versatile framework for quantitatively comparing structured metric spaces that arise in geometry and data analysis.

Abstract

We present theoretical properties of the space of metric pairs equipped with the Gromov--Hausdorff distance. First, we establish the classical metric separability and the geometric geodesicity of this space. Second, we prove an Arzelà--Ascoli-type theorem for metric pairs. Third, extending a result by Cassorla, we show that the set of pairs consisting of a $2$-dimensional compact Riemannian manifold and a $2$-dimensional submanifold with boundary that can be isometrically embedded in $\mathbb{R}^3$ is dense in the space of compact metric pairs. Finally, to broaden the scope of potential applications, we describe scenarios where the Gromov--Hausdorff distance between metric pairs or tuples naturally arises.

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Theorems & Definitions (38)

• Theorem \ref{thm:geodesic}
• Theorem \ref{thm:separable}
• Theorem \ref{thm:arzela}
• Theorem \ref{thm:cassorla}
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6