{
  "id": "1904.04065",
  "version": "v1",
  "published": "2019-04-05T10:00:30.000Z",
  "updated": "2019-04-05T10:00:30.000Z",
  "title": "On the Regions Formed by the Diagonals of a Convex Polygon",
  "authors": [
    "C P Anil Kumar"
  ],
  "comment": "25 pages, 7 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "For a positive integer $n\\geq 3$, the sides and diagonals of a convex $n$-gon divide the interior of the convex $n$-gon into finitely (polynomial in $n$) many regions bounded by them. Here in this article we associate to every region a unique $n$-cycle in the symmetric group $S_n$ of a certain type (defined as two standard consecutive cycle) by studying point arrangements in the plane. Then we find that there are more (exponential in $n$) number of such cycles leading to the conclusion that not every region labelled by a cycle appears in every convex $n$-gon. In fact most of them do not occur in any given single convex $n$-gon. Later in main Theorem $\\Omega$ of this article we characterize combinatorially those cycles (defined as definite cycles) whose corresponding regions occur in every convex $n$-gon and those cycles (defined as indefinite cycles) whose corresponding regions do not occur in every convex $n$-gon.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-05T10:00:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "51D20",
      "52C35"
    ],
    "keywords": [
      "convex polygon",
      "main theorem",
      "single convex",
      "corresponding regions occur",
      "indefinite cycles"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}