{
  "id": "1905.07988",
  "version": "v1",
  "published": "2019-05-20T10:53:27.000Z",
  "updated": "2019-05-20T10:53:27.000Z",
  "title": "Sharp estimate on the inner distance in planar domains",
  "authors": [
    "Danka Lučić",
    "Enrico Pasqualetto",
    "Tapio Rajala"
  ],
  "comment": "21 pages, 7 figures",
  "categories": [
    "math.MG",
    "math.CV"
  ],
  "abstract": "We show that the inner distance inside a bounded planar domain is at most the one-dimensional Hausdorff measure of the boundary of the domain. We prove this sharp result by establishing an improved Painlev\\'e length estimate for connected sets and by using the metric removability of totally disconnected sets, proven by Kalmykov, Kovalev, and Rajala. We also give a totally disconnected example showing that for general sets the Painlev\\'e length bound $\\kappa(E) \\le\\pi \\mathcal{H}^1(E)$ is sharp.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-20T10:53:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "28A75",
      "31A15"
    ],
    "keywords": [
      "sharp estimate",
      "one-dimensional hausdorff measure",
      "inner distance inside",
      "painleve length estimate",
      "painleve length bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}