{
  "abstract": "A classical theorem of Figiel asserts that every isometric embedding $φ\\colon X\\to Y$ between real Banach spaces, with $φ(0)=0$, admits a linear contractive left inverse on the closed linear span of $φ(X)$. We show that this phenomenon breaks down completely once local convexity is lost. More precisely, for every $0<p<1$ and every nonzero real Banach space $X$, we construct a $p$-Banach space $Y$ and an isometric embedding $φ\\colon X\\to Y$ such that $φ(0)=0$ and the closed linear span of $φ(X)$ is the entire space $Y$, but there is no linear map, bounded or not, on the linear span of $φ(X)$ which is a left inverse of $φ$. The construction is carried out using the $p$-Banach counterpart of Lipschitz-free spaces. In fact, $Y$ may be chosen isomorphic to the Lipschitz-free $p$-space over $X$. The main ingredient is a quantitative estimate controlling the distance to elementary molecules in Lipschitz-free $p$-spaces, a phenomenon specific to the range $0<p<1$. As a further consequence, we show that a real Banach space has the isometric $p$-lifting property if and only if its dimension is at most one.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07448",
  "author_total": 3,
  "authors": [
    {
      "name": "Fernando Albiac"
    },
    {
      "name": "José L. Ansorena"
    },
    {
      "name": "Marek Cúth"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0fnmk61s9c3ah.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07448",
  "fulltext_url": null,
  "id": "0fnmk61s9c3ah",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07448",
  "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": "W7221060588",
  "pdf_url": "https://arxiv.org/pdf/2610.07448",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0fnmk61s9c3ah.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07448",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07448"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Isometries without linear shadows: failure of Figiel's theorem beyond local convexity",
  "title_text": "Isometries without linear shadows: failure of Figiel's theorem beyond local convexity",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/0fnmk61s9c3ah",
  "venue": "arXiv",
  "versions": []
}