{
  "id": "0912.1356",
  "version": "v2",
  "published": "2009-12-07T22:05:36.000Z",
  "updated": "2012-01-06T23:20:31.000Z",
  "title": "How to take shortcuts in Euclidean space: making a given set into a short quasi-convex set",
  "authors": [
    "Jonas Azzam",
    "Raanan Schul"
  ],
  "comment": "Made referee edits",
  "doi": "10.1112/plms/pds005",
  "categories": [
    "math.MG",
    "math.CA"
  ],
  "abstract": "For a given connected set $\\Gamma$ in $d-$dimensional Euclidean space, we construct a connected set $\\tilde\\Gamma\\supset \\Gamma$ such that the two sets have comparable Hausdorff length, and the set $\\tilde\\Gamma$ has the property that it is quasiconvex, i.e. any two points $x$ and $y$ in $\\tilde\\Gamma$ can be connected via a path, all of which is in $\\tilde\\Gamma$, which has length bounded by a fixed constant multiple of the Euclidean distance between $x$ and $y$. Thus, for any set $K$ in $d-$dimensional Euclidean space we have a set $\\tilde\\Gamma$ as above such that $\\tilde\\Gamma$ has comparable Hausdorff length to a shortest connected set containing $K$. Constants appearing here depend only on the ambient dimension $d$. In the case where $\\Gamma$ is Reifenberg flat, our constants are also independent the dimension $d$, and in this case, our theorem holds for $\\Gamma$ in an infinite dimensional Hilbert space. This work closely related to $k-$spanners, which appear in computer science. Keywords: chord-arc, quasiconvex, k-spanner, traveling salesman.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-01-06T23:20:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75"
    ],
    "keywords": [
      "short quasi-convex set",
      "dimensional euclidean space",
      "connected set",
      "comparable hausdorff length",
      "infinite dimensional hilbert space"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.1356A"
    }
  }
}