{
  "abstract": "We study a factorization notion for Lipschitz functions between metric spaces in which such a function is written as a composition of a Hölder function and a function that suitably ``undoes'' the Hölder regularity. We show simple ways to construct such ``de-Höldering'' factorizations. If the identity mapping on a metric space $\\mathcal{M}$ admits a de-Höldering factorization through a metric space $\\mathcal{Z}$ that has a conical geodesic bicombing, then the class of metric spaces from which one can extend $\\mathcal{Z}$-valued Lipschitz functions is shown to be contained in the corresponding class of metric spaces for $\\mathcal{M}$-valued Lipschitz functions. As a quick consequence of these abstract permanence properties, we deduce that every $L_1$-valued Lipschitz function from a subset of $\\ell_2$ can be extended to a Lipschitz function that takes values in $L_1$ and is defined on all of $\\ell_2$, answering a 1992 question of Ball. By work of Makarychev and Makarychev, this implies that every weighted graph has a vertex cut sparsifier of size $n$ and quality $O(\\sqrt{\\log n})$, improving Moitra's 2009 bound. We also show that for every metric space $\\mathcal{Z}$ that has a conical geodesic bicombing, any metric transform of a metric space $\\mathcal{M}$ has $\\mathcal{Z}$-valued Lipschitz extension modulus at most a universal constant multiple of the $\\mathcal{Z}$-valued Lipschitz extension modulus of $\\mathcal{M}$ itself, improving the 2002 bound of Brudnyi and Shvartsman.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07564",
  "author_total": 1,
  "authors": [
    {
      "name": "Assaf Naor"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7yrbms9hejbq3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07564",
  "fulltext_url": null,
  "id": "7yrbms9hejbq3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07564",
  "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",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7211987019",
  "pdf_url": "https://arxiv.org/pdf/2609.07564",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7yrbms9hejbq3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:22Z",
      "native_id": "2609.07564",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07564"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "De-Höldering factorization",
  "title_text": "De-Höldering factorization",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/7yrbms9hejbq3",
  "venue": "arXiv",
  "versions": []
}