Gromov-Hausdorff versus Hausdorff distances for Euclidean submanifolds with positive reach

Abstract

This paper is about comparing the Gromov-Hausdorff distance between a pair of sets to the Hausdorff distance between the pair. Recent work on this problem has focused on two settings, Riemannian manifolds and metric graphs. Here, we focus on embedded submanifolds of Euclidean space endowed with the Euclidean metric. We prove that when $\mathcal{M}\subset\mathbb{R}^n$ is a compact connected embedded submanifold without boundary, with positive reach, and $\mathrm{A}\subset\mathcal{M}$ is a subset that is sufficiently dense in $\mathcal{M}$ with respect to the reach, the Gromov-Hausdorff distance between $\mathrm{A}$ and $\mathcal{M}$ is bounded below by $1/2$ of the Hausdorff distance between them. We extend this result in two ways. First, we show that the factor $1/2$ can be improved using Jung's theorem. Second, we prove bounds on the Gromov-Hausdorff distance for pairs of subsets of $\mathcal{M}$. When $\mathbb{S}^{1}$ is equipped with the two-dimensional Euclidean metric, the Gromov-Hausdorff distance between $\mathbb{S}^{1}$ and any sufficiently dense subset of $\mathbb{S}^{1}$ can be expressed explicitly as a function of the Hausdorff distance between the two. We use interleavings between geometric simplicial complexes, nerve lemmas, properties of reach and the nearest-point projection map, and the fundamental class to produce the inequalities for general $\mathcal{M}$. We use the Borsuk-Ulam theorem for $\mathcal{M}=\mathbb{S}^1\subset \mathbb{R}^2$.