{
  "id": "1209.4652",
  "version": "v2",
  "published": "2012-09-20T20:00:59.000Z",
  "updated": "2014-03-26T20:58:01.000Z",
  "title": "On the sum of the Voronoi polytope of a lattice with a zonotope",
  "authors": [
    "Mathieu Dutour Sikiric",
    "Viatcheslav Grishukhin",
    "Alexander Magazinov"
  ],
  "comment": "30 pages, 4 figures, 4 tables",
  "categories": [
    "math.MG",
    "math.CO"
  ],
  "abstract": "A parallelotope $P$ is a polytope that admits a facet-to-facet tiling of space by translation copies of $P$ along a lattice. The Voronoi cell $P_V(L)$ of a lattice $L$ is an example of a parallelotope. A parallelotope can be uniquely decomposed as the Minkowski sum of a zone closed parallelotope $P$ and a zonotope $Z(U)$, where $U$ is the set of vectors used to generate the zonotope. In this paper we consider the related question: When is the Minkowski sum of a general parallelotope and a zonotope $P+Z(U)$ a parallelotope? We give two necessary conditions and show that the vectors $U$ have to be free. Given a set $U$ of free vectors, we give several methods for checking if $P + Z(U)$ is a parallelotope. Using this we classify such zonotopes for some highly symmetric lattices. In the case of the root lattice $\\mathsf{E}_6$, it is possible to give a more geometric description of the admissible sets of vectors $U$. We found that the set of admissible vectors, called free vectors, is described by the well-known configuration of $27$ lines in a cubic. Based on a detailed study of the geometry of $P_V(\\mathsf{e}_6)$, we give a simple characterization of the configurations of vectors $U$ such that $P_V(\\mathsf{E}_6) + Z(U)$ is a parallelotope. The enumeration yields $10$ maximal families of vectors, which are presented by their description as regular matroids.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-03-26T20:58:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "voronoi polytope",
      "free vectors",
      "minkowski sum",
      "root lattice",
      "zone closed parallelotope"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.4652D"
    }
  }
}