Mathieu Dutour
Hebrew University of Jerusalem
Abstract:
A lattice is a rank n subgroup of R^n. A covering of R^n is a family of balls of equal radius such that any point belongs to at least one ball. The covering density is the average number of balls to which points of R^n belongs to. Our main purpose is to minimize the covering density in the lattice case: coverings defined by balls whose center belong to a lattice.
To any lattice L, one associates a Gram matrix G by taking a basis of the lattice. This is the key idea of Lattice Theory allowing to use analytic tools. A Delaunay polytope of a lattice is the convex hull of points lying on an empty sphere. They form a normal tesselation of R^n (dual to Voronoi tiling). The covering density is expressed in terms of maximum radius of Delaunay polytopes and determinant of the Gram matrix.
A L-type is defined as the set of matrices having the same Delaunay tesselation. This parameter space, together with semidefinite programming allows to solve the lattice covering problem, provided that one knows all the L-type domain. In practice, this is possible only up to dimension 5.
We will present the generalization of L-type theory to lattices having a fixed symmetry group. This will allow us to find best known covering in dimension 9-15.