Identify non-graph classes with equality between Gromov–Hausdorff and Hausdorff distances for dense samples

Identify classes of compact metric spaces beyond metric graphs for which there exists a density threshold ε > 0 (in terms of the Hausdorff distance to the subset) such that for any subset X whose Hausdorff distance to the ambient space is less than ε, the equality d_GH(M,X) = d_H(M,X) holds.

Background

In the graph setting, the authors provide conditions ensuring d_GH(G,X) ≥ d_H(G,X), yielding equality when the Gromov–Hausdorff distance is sufficiently small relative to graph geometry. Extending such equalities to other metric classes would enhance understanding of when GH and Hausdorff distances coincide for dense samples.

This problem seeks a structural characterization of metric spaces (beyond graphs) where sufficiently dense sampling guarantees GH–Hausdorff equality, potentially identifying new families of spaces amenable to similar analysis.