Let M be a closed Riemannian manifold and let \(X\subseteq M\) . If the sample X is sufficiently dense relative to the curvature of M, then the Gromov–Hausdorff distance between X and M is bounded from below by half their Hausdorff distance, namely \(d_\textrm{GH}(X,M) \ge \tfrac{1}{2} d_\textrm{H}(X,M)\) . The constant \(\tfrac{1}{2}\) can be improved depending on the dimension and curvature of the manifold M, and obtains the optimal value 1 in the case of the unit circle, meaning that if \(X\subseteq S^1\) satisfies \(d_\textrm{GH}(X,S^1)<\tfrac{\pi }{6}\) , then \(d_\textrm{GH}(X,S^1)=d_\textrm{H}(X,S^1)\) . We also provide versions lower bounding the Gromov–Hausdorff distance \(d_\textrm{GH}(X,Y)\) between two subsets \(X,Y\subseteq M\) . Our proofs convert discontinuous functions between metric spaces into simplicial maps between Čech or Vietoris–Rips complexes. We then produce topological obstructions to the existence of certain maps using the nerve lemma and the fundamental class of the manifold, thus lower bounding the Gromov–Hausdorff distance.
