{
  "abstract": "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12625",
  "author_total": 1,
  "authors": [
    {
      "name": "Paul Schott"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/305zbjebyrgz2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12625",
  "fulltext_url": null,
  "id": "305zbjebyrgz2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12625",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7213074491",
  "pdf_url": "https://arxiv.org/pdf/2609.12625",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/305zbjebyrgz2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12625",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12625"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "A converse bound for dGH vs. dH for Euclidean space and Riemannian manifolds",
  "title_text": "A converse bound for dGH vs. dH for Euclidean space and Riemannian manifolds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/305zbjebyrgz2",
  "venue": "arXiv",
  "versions": []
}