{
  "abstract": "We study almost Lipschitz embeddings in the quantitative geometry of unit spheres of classical \\(L_p\\)-spaces. Our main result is a sphere version of the classical Banach--Kadec--Paley theorem: for \\(1\\le p,q<\\infty\\), the unit sphere \\(S_{L_q}\\) admits an almost Lipschitz embedding into \\(L_p\\) if and only if \\(L_q\\) admits a linear embedding into \\(L_p\\). Consequently, the almost Lipschitz geometry of \\(L_p\\)-spheres completely determines the underlying linear structure of \\(L_p\\)-spaces. The proof relies on quantitative estimates for moduli of continuity of sphere mappings. These estimates are derived from Kalton--Randrianarivony's concentration inequalities, Mendel--Naor's metric cotype inequality, and Naor's sharp metric \\(X_p\\) inequality. They also yield several sharp results on the Hölder geometry of \\(L_p\\)-spheres.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.14641",
  "author_total": 3,
  "authors": [
    {
      "name": "Qingjin Cheng"
    },
    {
      "name": "Wuyi He"
    },
    {
      "name": "Bo Xiang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cj0pjkpc7y6qy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.14641",
  "fulltext_url": null,
  "id": "cj0pjkpc7y6qy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.14641",
  "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.FA"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7213273699",
  "pdf_url": "https://arxiv.org/pdf/2609.14641",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cj0pjkpc7y6qy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.14641",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.14641"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sphere version of Banach--Kadec--Paley theorem",
  "title_text": "Sphere version of Banach--Kadec--Paley theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/cj0pjkpc7y6qy",
  "venue": "arXiv",
  "versions": []
}