A converse bound for dGH vs. dH for Euclidean space and Riemannian manifolds

Abstract

We prove for the Euclidean space $\left({\mathbb{R}}^n,\Vert .{\Vert }_2\right)$ the existence of subsets $A^n\subseteq {\mathbb{R}}^n$ with $d_H({\mathbb{R}}^n,A^n)$ arbitrarily small, such that $d_{GH}({\mathbb{R}}^n,A^n)=\sqrt{\frac{n+1}{2n}}{d}_H({\mathbb{R}}^n,A^n)$, showing that the lower bound $d_{GH}({\mathbb{R}}^n,A^n)\ge \sqrt{\frac{n+1}{2n}}{d}_H({\mathbb{R}}^n,A^n)$ proved in arXiv:2607.18447 is tight for sets that are arbitrarily dense in the surrounding space. We also prove a similar statement for Riemannian manifolds and sufficiently dense subsets as a converse to arXiv:2309.16648.