Hausdorff vs Gromov-Hausdorff distances (2/4/2024)

Though Gromov-Hausdorff distances between metric spaces are a common concept in geometry and data analysis, these distances are hard to compute. If X and is a sufficiently dense subset of a closed Riemannian manifold M, then we lower bound the Gromov-Hausdorff distance between X and M by 1/2 the Hausdorff distance between them. The constant 1/2 can be improved depending on the dimension and curvature of the manifold, and obtains the optimal value 1 in the case of the circle. We also provide versions lower bounding the Gromov-Hausdorff distance between two subsets X and Y of the manifold M. Our proofs begin by converting discontinuous functions between metric spaces into simplicial maps between Čech complexes. We then produce topological obstructions to the existence of such maps using the nerve lemma and the fundamental class of the manifold. Joint with Florian Frick, Sushovan Majhi, Nicholas McBride.

[bennoarchimboldi]

Cool I’ll read this paper in a few years when I gain more mathematical maturity.