{
  "id": "2102.02037",
  "version": "v1",
  "published": "2021-02-03T12:44:09.000Z",
  "updated": "2021-02-03T12:44:09.000Z",
  "title": "The isometry group of Wasserstein spaces: the Hilbertian case",
  "authors": [
    "György Pál Gehér",
    "Tamás Titkos",
    "Dániel Virosztek"
  ],
  "comment": "30 pages, 2 figures",
  "categories": [
    "math.MG",
    "math-ph",
    "math.FA",
    "math.MP",
    "math.PR"
  ],
  "abstract": "Motivated by Kloeckner's result on the isometry group of the quadratic Wasserstein space $\\mathcal{W}_2\\left(\\mathbb{R}^n\\right)$, we describe the isometry group $\\mathrm{Isom}\\left(\\mathcal{W}_p (E)\\right)$ for all parameters $0 < p < \\infty$ and for all separable real Hilbert spaces $E.$ In fact, the $0<p<1$ case is a consequence of our more general result: we prove that $\\mathcal{W}_1(X)$ is isometrically rigid if $X$ is a complete separable metric space that satisfies the strict triangle inequality. Furthermore, we show that this latter rigidity result does not generalise to parameters $p>1$, by solving Kloeckner's problem affirmatively on the existence of mass-splitting isometries.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-03T12:44:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54E40",
      "46E27",
      "60A10",
      "60B05"
    ],
    "keywords": [
      "isometry group",
      "hilbertian case",
      "separable real hilbert spaces",
      "quadratic wasserstein space",
      "strict triangle inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}