Hausdorff vs Gromov-Hausdorff distances

TL;DR

Addresses how the Gromov--Hausdorff distance between a dense sampling $X$ and a closed Riemannian manifold $M$ relates to the Hausdorff distance, establishing a baseline lower bound $d_{ ext{GH}}(X,M) \ge \tfrac{1}{2}d_{ ext{H}}(X,M)$ under density assumptions and sharpening constants in special cases. The authors develop a framework that converts distortion bounds into continuous maps between Čech and Vietoris--Rips complexes, then uses the nerve lemma and the manifold's fundamental class to obstruct low-distortion embeddings. They further refine the constants via the filling radius, Jung's theorem, and curvature bounds, and obtain a sharp circle case where $d_{ ext{GH}}(X,S^1)=d_{ ext{H}}(X,S^1)$ for $d_{ ext{H}}(X,S^1)<\tfrac{\pi}{6}$. The paper also proves a two-subset bound, and shows that without density assumptions the ratio $d_{ ext{GH}}/d_{ ext{H}}$ can be arbitrarily small, highlighting intrinsic limitations and potential computational implications of Hausdorff-based estimates.

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 $d_{GH}(X,S^1)<\tfracπ{6}$, 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 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.

Figures (8)

• Figure 1: A manifold $M$, a subset $X$ (orange diamonds), and a subset $Y$ (blue circles).
• Figure 2: A finite set $X$ of points in the plane, along with three Čech complexes $\mathrm{\check{C}}\left(X;r\right)$ with increasing values of $r>0$.
• Figure 3: (Left) A good cover of a topological space and its nerve. (Right) A cover of a topological space that is not good, along with its nerve.
• Figure 4: Picture of the balls determining $\mathrm{\check{C}}\left(X;r+\varepsilon\right)$ with $r+\varepsilon < d_\mathrm{H}(X,M)$ in the proof of Theorem \ref{['thm:main1']}(a); note that the balls do not cover $M$.
• Figure 5: A finite set $X$ of points in the plane, along with three Vietoris--Rips complexes $\mathrm{VR}\left(X;r\right)$ with increasing values of $r>0$.

Theorems & Definitions (44)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 2.1: Correspondences and ambient Čech complexes ChazalDeSilvaOudot2014
• proof
• Definition 2.2: Convexity radius
• Lemma 2.3: Nerve lemma Alexandroff1928Borsuk1948Hatcher
• Corollary 2.4