The Gromov-Hausdorff distance, a dissimilarity measure between metric spaces, is used in computational topology and geometry to compare datasets that can be represented as metric spaces. Despite the computational obstructions to its practical use, it still provides a theoretical framework to quantify invariants’ stability and information loss. In this talk, we focus on a particular problem regarding the Gromov-Hausdorff distance: Given an object and a sample of it, under what conditions do their Hausdorff and Gromov-Hausdorff distances coincide? As the Gromov-Hausdorff distance describes how far they are from being isometric, and the Hausdorff distance measures the density of the sample, we can less formally restate the question as follows: When is a sample dense enough to describe the original object’s geometry faithfully? In particular, we discuss the case of metric graphs providing both negative and positive results.

Co-authors: Henry Adams, Sushovan Majhi, Fedor Manin, and Žiga Virk. Link to the preprint: https://arxiv.org/abs/2411.09182.