{
  "id": "2403.09245",
  "version": "v1",
  "published": "2024-03-14T10:10:27.000Z",
  "updated": "2024-03-14T10:10:27.000Z",
  "title": "Conditional plasticity of the unit ball of the $\\ell_\\infty$-sum of finitely many strictly convex Banach spaces",
  "authors": [
    "Kaarel August Kurik"
  ],
  "comment": "10 pages",
  "categories": [
    "math.FA"
  ],
  "abstract": "We prove that for any $\\ell_\\infty$-sum $Z = \\bigoplus_{i \\in [n]} X_i$ of finitely many strictly convex Banach spaces $(X_i)_{i \\in [n]}$, an extremeness preserving 1-Lipschitz bijection $f\\colon B_Z \\to B_Z$ is an isometry, by constraining the componentwise behavior of the inverse $g=f^{-1}$ with a theorem admitting a graph-theoretic interpretation. We also show that if $X, Y$ are Banach spaces, then a bijective 1-Lipschitz non-isometry of type $B_X \\to B_Y$ can be used to construct a bijective 1-Lipschitz non-isometry of type $B_X' \\to B_X'$ for some Banach space $X'$, and that a homeomorphic 1-Lipschitz non-isometry of type $B_X \\to B_X$ restricts to a homeomorphic 1-Lipschitz non-isometry of type $B_S \\to B_S$ for some separable subspace $S \\leq X$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-14T10:10:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B20",
      "47H09",
      "05C69"
    ],
    "keywords": [
      "strictly convex banach spaces",
      "unit ball",
      "conditional plasticity",
      "non-isometry",
      "graph-theoretic interpretation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}