{
  "abstract": "What is the Gromov--Hausdorff distance between Euclidean unit balls of different dimensions, denoted by $d_\\gh(B^m,B^n)$, for $m>n$? Note that the lower bound coming from the stability of persistent homology is zero, since all balls possess identical (trivial) persistent homology. To establish non-trivial lower bounds, we exploit the Borsuk--Ulam theorem. For any $m > n \\ge 1$, we prove that $d_\\gh(B^m,B^n)\\ge \\tfrac{1}{1+α_n} > \\frac{1}{2}$, where we determine $α_n$ explicitly. We also prove that $d_\\gh(B^m,B^n)\\to 1$ as $m\\to \\infty$ and that $d_\\gh(B^m,B^n)<1$ for all finite $m>n\\geq 1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09652",
  "author_total": 2,
  "authors": [
    {
      "name": "Henry Adams"
    },
    {
      "name": "Kushagri Sharma"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5xbz46hje6b7j.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09652",
  "fulltext_url": null,
  "id": "5xbz46hje6b7j",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09652",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7212177442",
  "pdf_url": "https://arxiv.org/pdf/2609.09652",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5xbz46hje6b7j.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.09652",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09652"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Gromov--Hausdorff Distance Between Euclidean Unit Balls",
  "title_text": "Gromov--Hausdorff Distance Between Euclidean Unit Balls",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/5xbz46hje6b7j",
  "venue": "arXiv",
  "versions": []
}