{
  "abstract": "For $1\\le m<n$, let $c_{m,n}$ be the infimum of the scales $r$ at which the Vietoris-Rips simplicial complex of the unit round sphere $\\mathbb{S}^m$, equipped with its geodesic metric, admits a continuous odd map from $\\mathbb{S}^n$. Previous work established the lower bound $\\frac{1}{2}c_{m,n}\\le d_{\\mathrm{GH}}(\\mathbb{S}^m,\\mathbb{S}^n)$ through a quantitative Borsuk-Ulam theorem and raised the question of whether it is always sharp. We show that this bound can underestimate the distance by a factor arbitrarily close to two, the largest possible factor. Combining projective-code upper bounds for $c_{m,n}$ with lower bounds for $d_{\\mathrm{GH}}$ obtained from obstructions to homomorphisms between Borsuk graphs, we obtain explicit finite-dimensional bounds. Along sequences of pairs $1\\le m<n$, we have $c_{m,n}\\to\\fracπ{2}$ whenever $\\log n/m\\to0$, while $d_{\\mathrm{GH}}(\\mathbb{S}^m,\\mathbb{S}^n)\\to\\fracπ{2}$ whenever $m/n\\to0$. When both conditions hold (for example when $n=m^2$) the ratio of the distance to $\\frac{1}{2}c_{m,n}$ tends to two. For each fixed $m\\ge3$, the inequality is strict for all sufficiently large $n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07778",
  "author_total": 3,
  "authors": [
    {
      "name": "Sunhyuk Lim"
    },
    {
      "name": "Facundo Mémoli"
    },
    {
      "name": "Qingsong Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0d8w6r226ae56.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07778",
  "fulltext_url": null,
  "id": "0d8w6r226ae56",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07778",
  "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",
      "1703|Topological and Geometric Data Analysis"
    ]
  ],
  "openalex": "W7212076529",
  "pdf_url": "https://arxiv.org/pdf/2609.07778",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0d8w6r226ae56.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2609.07778",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07778"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Gromov-Hausdorff Distances Between Spheres: Asymptotics and Vietoris-Rips Coindex Bounds",
  "title_text": "Gromov-Hausdorff Distances Between Spheres: Asymptotics and Vietoris-Rips Coindex Bounds",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/0d8w6r226ae56",
  "venue": "arXiv",
  "versions": []
}