{
  "abstract": "We determine the asymptotic order of the observable diameter of $\\ell^p$-product spaces associated with a probability measure on $[0,1]$ whose density has a power-type interior zero. Different vanishing orders are allowed on the two sides of the zero. For every $1 \\le p \\le \\infty$, we obtain sharp two-sided estimates and show that the asymptotic order is governed by the larger of the two vanishing orders. In the case $p=2$, the transition occurs exactly at the threshold for the spectral gap.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35889",
  "author_total": 1,
  "authors": [
    {
      "name": "Takashi Shioya"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1p7y31e6jag78.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35889",
  "fulltext_url": null,
  "id": "1p7y31e6jag78",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35889",
  "language": null,
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  "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.FA"
    ],
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      "arxiv",
      "math.MG"
    ],
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      "arxiv",
      "math.PR"
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      "2610|Spectral Theory in Mathematical Physics"
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  "openalex": "W7224101077",
  "pdf_url": "https://arxiv.org/pdf/2610.35889",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1p7y31e6jag78.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.35889",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35889"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Observable diameter of product measures with a power-type interior zero",
  "title_text": "Observable diameter of product measures with a power-type interior zero",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/1p7y31e6jag78",
  "venue": "arXiv",
  "versions": []
}