engleski

Perfect forms and perfect Delaunay

A lattice packing is a family of non-overlapping balls whose center belongs to a lattice L. The density of a packing is the fraction of space occupied by them and the lattice packing problem is to find the lattice of maximum density. We will expose the theory of Voronoi for lattice packings, that is the notion of perfect form, Ryshkov polyhedron and Voronoi algorithm that allow to solve the lattice packing problem up to dimension 8. Along the way we will shortly introduce root lattices, the Leech lattice and the well rounded retract. Then we will consider the covering problem where one wants to cover the space by balls whose center belong to a lattice. We will shortly discussthe problem of minimizing the covering density and then we will remark that the root lattice E6 is actually a local maxima for the covering density. This allows us to introduce the Erdahl cone, the notion of perfect Delaunay polytopes and a Voronoi algorithm for their enumeration.

polytopes; lattices; sphere packing problem; sphere covering problem.

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