{
  "id": "math/0606371",
  "version": "v1",
  "published": "2006-06-15T17:28:29.000Z",
  "updated": "2006-06-15T17:28:29.000Z",
  "title": "On a Convex Operator for Finite Sets",
  "authors": [
    "Branko Ćurgus",
    "Krzysztof Kołodziejczyk"
  ],
  "comment": "20 pages, 16 figures",
  "journal": "Discrete Applied Mathematics 155 (2007) no. 13, 1774--1792",
  "categories": [
    "math.MG"
  ],
  "abstract": "Let $S$ be a finite set with $n$ elements in a real linear space. Let $\\cJ_S$ be a set of $n$ intervals in $\\nR$. We introduce a convex operator $\\co(S,\\cJ_S)$ which generalizes the familiar concepts of the convex hull $\\conv S$ and the affine hull $\\aff S$ of $S$. We establish basic properties of this operator. It is proved that each homothet of $\\conv S$ that is contained in $\\aff S$ can be obtained using this operator. A variety of convex subsets of $\\aff S$ can also be obtained. For example, this operator assigns a regular dodecagon to the 4-element set consisting of the vertices and the orthocenter of an equilateral triangle. For $\\cJ_S$ which consists of bounded intervals, we give the upper bound for the number of vertices of the polytope $\\co(S,\\cJ_S)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-06-15T17:28:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52A05"
    ],
    "keywords": [
      "finite set",
      "convex operator",
      "real linear space",
      "familiar concepts",
      "convex hull"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......6371C"
    }
  }
}