{
  "id": "2405.01860",
  "version": "v1",
  "published": "2024-05-03T05:22:07.000Z",
  "updated": "2024-05-03T05:22:07.000Z",
  "title": "Characterizing Lipschitz images of injective metric spaces",
  "authors": [
    "Judyta Bąk",
    "Taras Banakh",
    "Joanna Garbulińska-Węgrzyn",
    "Magdalena Nowak",
    "Michał Popławski"
  ],
  "categories": [
    "math.GN"
  ],
  "abstract": "A metric space $X$ is {\\em injective} if every non-expanding map $f:B\\to X$ defined on a subspace $B$ of a metric space $A$ can be extended to a non-expanding map $\\bar f:A\\to X$. We prove that a metric space $X$ is a Lipschitz image of an injective metric space if and only if $X$ is Lipschitz connected in the sense that for every points $x,y\\in X$, there exists a Lipschitz map $f:[0,1]\\to X$ such that $f(0)=x$ and $f(1)=y$. In this case the metric space $X$ carries a well-defined intrinsic metric. A metric space $X$ is a Lipschitz image of a compact injective metric space if and only if $X$ is compact, Lipschitz connected and its intrinsic metric is totally bounded. A metric space $X$ is a Lipschitz image of a separable injective metric space if and only if $X$ is a Lipschitz image of the Urysohn universal metric space if and only if $X$ is analytic, Lipschitz connected and its intrinsic metric is separable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-05-03T05:22:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "54E35",
      "54E40",
      "51F30",
      "54C55",
      "54E45",
      "54E50",
      "54F15"
    ],
    "keywords": [
      "characterizing lipschitz images",
      "urysohn universal metric space",
      "non-expanding map",
      "compact injective metric space",
      "separable injective metric space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}