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

RESUMO

We prove for the Euclidean space $\left(\mathbb{R}^n, \|.\|_2\right)$ 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 $d_{GH}(\mathbb{R}^n, A^n) \geq \sqrt{\frac{n+1}{2n}}d_H(\mathbb{R}^n, A^n)$ 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 .