{
  "abstract": "Let $X$, $Y$ be real Banach spaces with closed unit balls $B_X$, $B_Y$ and unit spheres $S_X$, $S_Y$. We prove that every non-expansive map $F\\colon B_X\\to B_Y$ with $F(S_X)\\subseteq S_Y\\subseteq F(B_X)$ is the restriction of a surjective linear isometry $X\\to Y$. Consequently, a non-expansive bijection of $B_X$ onto itself that fails to be an isometry must map some point of $S_X$ into the open unit ball, so the plasticity of $B_X$ reduces to showing that non-expansive bijections preserve norm 1. The proof shows that $F$ is Fréchet differentiable at $0$ with a surjective linear isometry $T$ as its derivative, and then proves $T^{-1}F=\\mathrm{id}$ using a quantitative rigidity lemma for non-expansive self-maps $H$ of $B_X$ satisfying $\\|a+H(a)\\|=2$ for $a\\in S_X$; the latter relies on the Baillon-Bruck rate of asymptotic regularity for Krasnosel'skii-Mann iterations. As a corollary, the identity is extremal in the set of non-expansive self-maps of $B_X$ for every Banach space $X$, which together with a result of Bargetz, Dymond and Pirk shows that all surjective isometries of $B_X$ are extremal, answering a question of theirs, including the case of $L^1[0,1]$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.28111",
  "author_total": 1,
  "authors": [
    {
      "name": "Kaarel August Kurik"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bm70baaajv59g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.28111",
  "fulltext_url": null,
  "id": "bm70baaajv59g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.28111",
  "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",
      "math.FA"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7223110229",
  "pdf_url": "https://arxiv.org/pdf/2610.28111",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bm70baaajv59g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.28111",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.28111"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Non-expansive maps between unit balls that send the sphere onto the sphere are linear isometries",
  "title_text": "Non-expansive maps between unit balls that send the sphere onto the sphere are linear isometries",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/bm70baaajv59g",
  "venue": "arXiv",
  "versions": []
}