{
  "id": "1410.6430",
  "version": "v1",
  "published": "2014-10-23T17:48:14.000Z",
  "updated": "2014-10-23T17:48:14.000Z",
  "title": "Convex-normal (pairs of) polytopes",
  "authors": [
    "Christian Haase",
    "Jan Hofmann"
  ],
  "comment": "10 pages, 8 figures",
  "categories": [
    "math.CO",
    "math.AC",
    "math.AG"
  ],
  "abstract": "In 2012 Gubeladze (Adv.\\ Math.\\ 2012) introduced the notion of k-convex-normal polytopes to show that integral polytopes all of whose edges are longer than 4d(d+1) have the integer decomposition property. In the first part of this paper we show that for lattice polytopes there is no difference between k- and (k+1)-convex-normality (for k >= 3) and improve the bound to 2d(d+1). In the second part we extend the definition to pairs of polytopes and show that for rational polytopes P and Q, where the normal fan of P is a refinement of the normal fan of Q, if every edge e_P of P is at least d times as long as the corresponding edge e_Q of Q, then (P+Q) \\cap \\Z^d = (P\\cap \\Z^d) + (Q \\cap \\Z^d).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-23T17:48:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B20"
    ],
    "keywords": [
      "normal fan",
      "integer decomposition property",
      "second part",
      "integral polytopes",
      "lattice polytopes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.6430H"
    }
  }
}