{
  "abstract": "The problem of differentiating Lipschitz curves in Banach spaces goes back to Tamarkin in the early 1930s and is one of the origins of the Radon--Nikodým property in classical Banach space theory. In this article we characterize those quasi-Banach spaces $X$ for which every Lipschitz curve $F\\to\\mathbb{R}\\to X$ admits a point of differentiability and prove that this property holds if and only if $X$ is isomorphic to a Banach space with the Radon--Nikodým property, thus substantiating a conjecture of Kalton, communicated personally to the first-named author in 2008. Equivalently, every nonlocally convex quasi-Banach space admits a nowhere differentiable Lipschitz curve. Our result is obtained from a quantitative characterization of local convexity in terms of the infinitesimal oscillation of Lipschitz curves. Motivated by this obstruction, we then investigate positive differentiability phenomena in $\\ell_p$, $0<p<1$, and identify natural decoupled-coordinate constructions for which Lipschitz regularity nevertheless implies almost-everywhere differentiability. We also contrast this behavior with metric differentiability for the natural metric and the quasi-metric on $\\ell_p$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26992",
  "author_total": 3,
  "authors": [
    {
      "name": "Fernando Albiac"
    },
    {
      "name": "José L. Ansorena"
    },
    {
      "name": "Pietro Wald"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5cegqha5nqdyv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26992",
  "fulltext_url": null,
  "id": "5cegqha5nqdyv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26992",
  "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": "W7214198690",
  "pdf_url": "https://arxiv.org/pdf/2609.26992",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5cegqha5nqdyv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.26992",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26992"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Differentiability of Lipschitz curves in $p$-Banach spaces",
  "title_text": "Differentiability of Lipschitz curves in p-Banach spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/5cegqha5nqdyv",
  "venue": "arXiv",
  "versions": []
}