Mathieu Dutour Sikirić

COMPLEXITY AND ALGORITHMS FOR COMPUTING VORONOI CELLS OF LATTICES

In this paper we are concerned with finding the vertices of the Voronoi cell of a Euclidean lattice. Given a basis of a lattice, we prove that computing the number of vertices is a #P-hard problem. On the other hand we describe an algorithm for this problem which is especially suited for low dimensional (say dimensions at most 12) and for highly-symmetric lattices. We use our implementation, which drastically outperforms those of current computer algebra systems, to find the vertices of Voronoi cells and quantizer constants of some prominent lattices.

Classification of eight-dimensional perfect forms

In this paper, we classify the perfect lattices in dimension 8. There are 10916 of them. Our classification heavily relies on exploiting symmetry in polyhedral computations. Here we describe algorithms making the classification possible.

Cube packings, second moment and holes

We consider tilings and packings of R^d by integral translates of cubes [0,2[^d, which are 4Z^d-periodic. Such cube packings can be described by cliques of an associated graph, which allow us to classify them in dimensions [emailxa0protected]?4. For higher dimensions, we use random methods for generating some examples. Such a cube packing is called non-extendible if we cannot insert a cube in the complement of the packing. In dimension 3, there is a unique non-extendible cube packing with 4 cubes. We prove that d-dimensional cube packings with more than 2^d-3 cubes can be extended to cube tilings. We also give a lower bound on the number N of cubes of non-extendible cube packings. Given such a cube packing and [emailxa0protected]?Z^d, we denote by Nz the number of cubes inside the 4-cube z+[0,4[^d and call the second moment the average of Nz^2. We prove that the regular tiling by cubes has maximal second moment and gives a lower bound on the second moment of a cube packing in terms of its density and dimension.

Computing symmetry groups of polyhedra

Knowing the symmetries of a polyhedron can be very useful for the analysis of its structure as well as for practical polyhedral computations. In this note, we study symmetry groups preserving the linear, projective and combinatorial structure of a polyhedron. In each case we give algorithmic methods to compute the corresponding group and discuss some practical experiences. For practical purposes the linear symmetry group is the most important, as its computation can be directly translated into a graph automorphism problem. We indicate how to compute integral subgroups of the linear symmetry group that are used, for instance, in integer linear programming.

Combinatorial cube packings in the cube and the torus

We consider sequential random packing of cubes

Random sequential packing of cubes

z+[0,1]^n

On the integral homology of PSL4(Z) and other arithmetic groups

with

The Contact Polytope of the Leech Lattice

zin frac{1}{N}ZZ^n

How to compute the rank of a Delaunay polytope

into the cube

Inhomogeneous extreme forms

[0,2]^n