Hausdorff vs Gromov-Hausdorff distances

Abstract

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 $d_{GH}(X,M) \ge \frac{1}{2} d_H(X,M)$. The constant $\frac{1}{2}$ 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 $d_{GH}(X,S^1)<\tfrac{\pi}{6}$, then $d_{GH}(X,S^1)=d_H(X,S^1)$. We also provide versions lower bounding the Gromov-Hausdorff distance $d_{GH}(X,Y)$ between two subsets $X,Y\subseteq M$. Our proofs convert discontinuous functions between metric spaces into simplicial maps between \v{C}ech 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.