{
  "abstract": "In a high-dimensional Funk ball, the endpoint potential makes the forward and reverse distances behave differently, while symmetrization erases the distinction that survives in pyramid limits. For Euclidean beta-type radial measures, we determine the natural-scale limits in all three parameter phases. The divergent phase yields directed Gaussian pyramids indexed by the balance between dimension and radial concentration. A positive limiting parameter yields a Gaussian--chi Funk horocone pyramid, built from Gaussian bases, chi-distributed heights, and a directed endpoint-potential increment, while a vanishing parameter yields the universal directed star pyramid. In the last phase, the dimension-dependent logarithmic contributions cancel exactly, so the conclusion requires no further rate condition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27752",
  "author_total": 1,
  "authors": [
    {
      "name": "Shigeaki Yokota"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/616gbecsqa9er.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27752",
  "fulltext_url": null,
  "id": "616gbecsqa9er",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27752",
  "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.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7214195828",
  "pdf_url": "https://arxiv.org/pdf/2609.27752",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/616gbecsqa9er.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27752",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27752"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Funk Beta Balls: Exact Potentials and a Three-Phase Pyramid Diagram",
  "title_text": "Funk Beta Balls: Exact Potentials and a Three-Phase Pyramid Diagram",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/616gbecsqa9er",
  "venue": "arXiv",
  "versions": []
}