{
  "id": "1110.6124",
  "version": "v2",
  "published": "2011-10-27T15:49:41.000Z",
  "updated": "2011-10-28T13:59:25.000Z",
  "title": "A planar bi-Lipschitz extension Theorem",
  "authors": [
    "Sara Daneri",
    "Aldo Pratelli"
  ],
  "comment": "55 pages, 21 figures",
  "categories": [
    "math.FA",
    "math.AP"
  ],
  "abstract": "We prove that, given a planar bi-Lipschitz homeomorphism $u$ defined on the boundary of the unit square, it is possible to extend it to a function $v$ of the whole square, in such a way that $v$ is still bi-Lipschitz. In particular, denoting by $L$ and $\\widetilde L$ the bi-Lipschitz constants of $u$ and $v$, with our construction one has $\\widetilde L \\leq C L^4$ (being $C$ an explicit geometrical constant). The same result was proved in 1980 by Tukia (see \\cite{Tukia}), using a completely different argument, but without any estimate on the constant $\\widetilde L$. In particular, the function $v$ can be taken either smooth or (countably) piecewise affine.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-10-28T13:59:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "planar bi-lipschitz extension theorem",
      "planar bi-lipschitz homeomorphism",
      "unit square",
      "bi-lipschitz constants",
      "explicit geometrical constant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.6124D"
    }
  }
}