{
  "id": "1911.03767",
  "version": "v1",
  "published": "2019-11-09T20:04:14.000Z",
  "updated": "2019-11-09T20:04:14.000Z",
  "title": "Any isometry between the spheres of sufficiently smooth $2$-dimensional Banach spaces is linear",
  "authors": [
    "Taras Banakh"
  ],
  "comment": "20 pages",
  "categories": [
    "math.FA",
    "math.CA",
    "math.DG",
    "math.MG"
  ],
  "abstract": "A $2$-dimensional Banach space $X$ is called $\\breve W{}^{3,1}$-smooth if the polar parametrization $\\mathbf p:\\mathbb R\\to S_X$ of its unit sphere $S_X=\\{x\\in X:\\|x\\|=1\\}$ has locally absolutely continuous second derivative $\\mathbf p''$ and the vector $\\mathbf p''(t)$ is not collinear to $\\mathbf p'(t)$ for almost all $t\\in\\mathbb R$. We prove that any isometry $f:S_X\\to S_Y$ between the unit spheres of $S\\breve W{}^{3,1}$-smooth $2$-dimensional Banach spaces $X,Y$ extends to a linear isometry $\\bar f:X\\to Y$ of the Banach spaces $X,Y$. This answers the famous Tingley's problem in the class of $\\breve W{}^{3,1}$-smooth $2$-dimensional Banach spaces.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-09T20:04:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B04",
      "46B20",
      "53A04",
      "26A46",
      "26A24",
      "46E35"
    ],
    "keywords": [
      "dimensional banach space",
      "sufficiently smooth",
      "unit sphere",
      "absolutely continuous second derivative",
      "polar parametrization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}