When the Gromov-Hausdorff distance between finite-dimensional space and its subset is finite?

2024/11/20 by I. N. Mikhailov, Mikhailov, I. N., A. A. Tuzhilin +1
Mathematics · #46B20 #51F99 #Advanced Topology and Set Theory #FOS: Mathematics #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2411.13539

openalex publication_date 2024/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we prove that the Gromov--Hausdorff distance between ℝⁿ and its subset A is finite if and only if A is an ε-net in ℝⁿ for some ε>0. For infinite-dimensional Euclidean spaces this is not true. The proof is essentially based on upper estimate of the Euclidean Gromov--Hausdorff distance by means of the Gromov-Hausdorff distance.

Related

• Gromov-Hausdorff distances between normed spaces
• Gromov-Hausdorff distance and Jung constant of finite-dimensional normed spaces
• Calculating Gromov-Hausdorff distance by means of asymptotic dimension
• Fundamentals of Theory of Continuous Gromov--Hausdorff distance
• A direct proof of one Gromov's theorem