Metric pairs and tuples in theory and applications

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.