{
  "id": "1904.01343",
  "version": "v1",
  "published": "2019-04-02T11:30:39.000Z",
  "updated": "2019-04-02T11:30:39.000Z",
  "title": "Families of lattice polytopes of mixed degree one",
  "authors": [
    "Gabriele Balletti",
    "Christopher Borger"
  ],
  "comment": "14 pages, 3 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "It has been shown by Soprunov that the normalized mixed volume (minus one) of an $n$-tuple of $n$-dimensional lattice polytopes is a lower bound for the number of interior lattice points in the Minkowski sum of the polytopes. He defined $n$-tuples of mixed degree at most one to be exactly those for which this lower bound is attained with equality, and posed the problem of a classification of such tuples. We give a finiteness result regarding this problem in general dimension $n \\geq 4$, showing that all but finitely many $n$-tuples of mixed degree at most one admit a common lattice projection onto the unimodular simplex $\\Delta_{n-1}$. Furthermore, we give a complete solution in dimension $n=3$. In the course of this we show that our finiteness result does not extend to dimension $n=3$, as we describe infinite families of triples of mixed degree one not admitting a common lattice projection onto the unimodular triangle $\\Delta_2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-04-02T11:30:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "52B20",
      "52A39"
    ],
    "keywords": [
      "mixed degree",
      "common lattice projection",
      "finiteness result",
      "lower bound",
      "interior lattice points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}