Lipschitz (non-)equivalence of the Gromov$\unicode{x2013}$Hausdorff distances, including on ultrametric spaces

The Gromov$\unicode{x2013}$Hausdorff distance measures the difference in shape between compact metric spaces. While even approximating the distance up to any practical factor poses an NP-hard problem, its relaxations have proven useful for the problems in geometric data analysis, including on point clouds, manifolds, and graphs. We investigate the modified Gromov$\unicode{x2013}$Hausdorff distance, a relaxation of the standard distance that retains many of its theoretical properties, which includes their topological equivalence on a rich set of families of metric spaces. We show that the two distances are Lipschitz-equivalent on any family of metric spaces of uniformly bounded size, but that the equivalence does not hold in general, not even when the distances are restricted to ultrametric spaces. We additionally prove that the standard and the modified Gromov$\unicode{x2013}$Hausdorff distances are either equal or within a factor of 2 from each other when taken to a regular simplex, which connects the relaxation to some well-known problems in discrete geometry.