{
  "id": "1108.3895",
  "version": "v1",
  "published": "2011-08-19T05:22:18.000Z",
  "updated": "2011-08-19T05:22:18.000Z",
  "title": "Disjoint Empty Convex Pentagons in Planar Point Sets",
  "authors": [
    "Bhaswar B. Bhattacharya",
    "Sandip Das"
  ],
  "comment": "23 pages, 28 figures",
  "journal": "Periodica Mathematica Hungarica, Vol. 66 (1), 73--86, 2013",
  "categories": [
    "math.CO"
  ],
  "abstract": "Harborth [{\\it Elemente der Mathematik}, Vol. 33 (5), 116--118, 1978] proved that every set of 10 points in the plane, no three on a line, contains an empty convex pentagon. From this it follows that the number of disjoint empty convex pentagons in any set of $n$ points in the plane is least $\\lfloor\\frac{n}{10}\\rfloor$. In this paper we prove that every set of 19 points in the plane, no three on a line, contains two disjoint empty convex pentagons. We also show that any set of $2m+9$ points in the plane, where $m$ is a positive integer, can be subdivided into three disjoint convex regions, two of which contains $m$ points each, and another contains a set of 9 points containing an empty convex pentagon. Combining these two results, we obtain non-trivial lower bounds on the number of disjoint empty convex pentagons in planar points sets. We show that the number of disjoint empty convex pentagons in any set of $n$ points in the plane, no three on a line, is at least $\\lfloor\\frac{5n}{47}\\rfloor$. This bound has been further improved to $\\frac{3n-1}{28}$ for infinitely many $n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-08-19T05:22:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52C10",
      "52A10"
    ],
    "keywords": [
      "disjoint empty convex pentagons",
      "planar point sets",
      "planar points sets",
      "non-trivial lower bounds",
      "disjoint convex regions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.3895B"
    }
  }
}