{
  "id": "0907.0776",
  "version": "v1",
  "published": "2009-07-04T19:40:30.000Z",
  "updated": "2009-07-04T19:40:30.000Z",
  "title": "Delaunay polytopes derived from the Leech lattice",
  "authors": [
    "Mathieu Dutour Sikiric",
    "Konstantin Rybnikov"
  ],
  "comment": "16 pages, 1 table",
  "categories": [
    "math.NT",
    "math.GT"
  ],
  "abstract": "Given a lattice L of R^n, a polytope D is called a Delaunay polytope in L if the set of its vertices is S\\cap L where S is a sphere having no lattice points in its interior. D is called perfect if the only ellipsoid in R^n that contains S\\cap L is exactly S. For a vector v of the Leech lattice \\Lambda_{24} we define \\Lambda_{24}(v) to be the lattice of vectors of \\Lambda_{24} orthogonal to v. We studied Delaunay polytopes of L=\\Lambda_{24}(v) for |v|^2<=22. We found some remarkable examples of Delaunay polytopes in such lattices and disproved a number of long standing conjectures. In particular, we discovered: --Perfect Delaunay polytopes of lattice width 4; previously, the largest known width was 2. --Perfect Delaunay polytopes in L, which can be extended to perfect Delaunay polytopes in superlattices of L of the same dimension. --Polytopes that are perfect Delaunay with respect to two lattices $L\\subset L'$ of the same dimension. --Perfect Delaunay polytopes D for L with |Aut L|=6|Aut D|: all previously known examples had |Aut L|=|Aut D| or |Aut L|=2|Aut D|. --Antisymmetric perfect Delaunay polytopes in L, which cannot be extended to perfect (n+1)-dimensional centrally symmetric Delaunay polytopes. --Lattices, which have several orbits of non-isometric perfect Delaunay polytopes. Finally, we derived an upper bound for the covering radius of \\Lambda_{24}(v)^{*}, which generalizes the Smith bound and we prove that it is met only by \\Lambda_{23}^{*}, the best known lattice covering in R^{23}.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-07-04T19:40:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "leech lattice",
      "non-isometric perfect delaunay polytopes",
      "antisymmetric perfect delaunay polytopes",
      "centrally symmetric delaunay polytopes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.0776D"
    }
  }
}