Mathematics Colloquia and Seminars

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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 tessellation 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.

L-type were introduced by Voronoi and are defined as the set of Gram matrices having the same Delaunay tessellation. This parameter space, together with a semidefinite programming program of Vallentin and Schuermann allow us to solve the lattice covering problem, provided that one knows all L-type domains. 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.

Then, we will consider the following extensions of the theory:

• to the case of describing a single Delaunay polytope in a lattice.
• to the case of several orbits of points under translation in searching for non-lattice coverings.

We will mention in passing many interesting and "record breaking" structures that show up in this work.