{
  "abstract": "Kimberling's Problem 10 asks for a simple closed curve of prescribed length $L$ (in particular, $L=4π$) on the unit sphere minimizing the mean geodesic distance $\\mathcal{J}$ from a point of the sphere to the curve. For a positive integer $n$, put $\\vartheta_{n}=π/(2n)$ and $L_{n}=2π/\\sin\\vartheta_{n}$. We show that the minimum of $\\mathcal{J}$ over rectifiable simple closed curves of length at most $L_{n}$ equals $\\vartheta_{n}-\\tan(\\vartheta_{n}/2)$, that it is attained only by curves of length exactly $L_{n}$, and that the sphere-filling ropes $β^{n,k}$ of Gerlach and von der Mosel attain it. Kimberling's case is $n=3$: at $L=4π$ the minimum is $π/6+\\sqrt{3}-2=0.255649\\ldots$, attained by an explicit six-arc curve and by its mirror image. For $L\\le2π$ we determine $J(L)$, the infimum of $\\mathcal{J}$ over curves of length $L$, exactly: it equals $π/2-L/(2π)$, attained precisely by the circles of length $L$. At the lengths $L_{n}$ we do not classify all minimizers, but show that every one of them bisects the sphere into two disks of area $2π$ and inradius $\\vartheta_{n}$ whose inward collars have the largest possible area at every depth. The great circle is the only minimizer for $n=1$, and the $β^{n,k}$ are, up to congruence, the only ones of thickness at least $\\sin\\vartheta_{n}$. For arbitrary $L$ the function $J$ is nonincreasing, and together with the above this brackets it between two explicit values.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09638",
  "author_total": 1,
  "authors": [
    {
      "name": "J. V. M. Pimentel"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ft9pd2qw9k7n7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09638",
  "fulltext_url": null,
  "id": "ft9pd2qw9k7n7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09638",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DG"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7212153240",
  "pdf_url": "https://arxiv.org/pdf/2609.09638",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ft9pd2qw9k7n7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09638",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09638"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The mean distance to a simple closed curve on the sphere",
  "title_text": "The mean distance to a simple closed curve on the sphere",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/ft9pd2qw9k7n7",
  "venue": "arXiv",
  "versions": []
}