{
  "id": "2411.13539",
  "version": "v1",
  "published": "2024-11-20T18:42:15.000Z",
  "updated": "2024-11-20T18:42:15.000Z",
  "title": "When the Gromov-Hausdorff distance between finite-dimensional space and its subset is finite?",
  "authors": [
    "I. N. Mikhailov",
    "A. A. Tuzhilin"
  ],
  "comment": "6 pages, accepted for publication in Commun. Math. Res. (http://www.global-sci.org/intro/online.html?journal=cmr)",
  "categories": [
    "math.MG"
  ],
  "abstract": "In this paper we prove that the Gromov--Hausdorff distance between $\\mathbb{R}^n$ and its subset $A$ is finite if and only if $A$ is an $\\varepsilon$-net in $\\mathbb{R}^n$ for some $\\varepsilon>0$. 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-20T18:42:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B20",
      "51F99"
    ],
    "keywords": [
      "finite-dimensional space",
      "euclidean gromov-hausdorff distance",
      "infinite-dimensional euclidean spaces",
      "upper estimate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}