Metric pairs and tuples in theory and applications

TL;DR

This paper explores the mathematical properties of the space of metric pairs under the Gromov--Hausdorff distance, establishing foundational results and demonstrating the density of certain geometric pairs within this space.

Contribution

It provides new theoretical insights into the structure of metric pairs, including separability, geodesicity, an Arzelà--Ascoli-type theorem, and density results for specific geometric pairs.

Findings

The space of metric pairs is metrically separable.

The space of metric pairs is geometrically geodesic.

Certain geometric pairs are dense in the space of metric pairs.

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 Arzel\`a--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.