0 Quadratic forms of rank 1 , closed zones and laminae

For a given lattice, we establish an equivalence involving a closed zone of the corresponding Voronoi polytope, a lamina hyperplane of the corresponding Delaunay partition and a quadratic form of rank 1 being an extreme ray of the corresponding L-type domain. 1991 Mathematics Subject classification: primary 52C07; secondary 11H55 An n-dimensional lattice determines two normal partitions of the n space R n into polytopes. These are the Voronoi partition and the Delaunay partition. These partitions are combinatorially dual, i.e. a k-dimensional face of one partition is orthogonal to an (n − k)-dimensional face of the other partition. Besides, a vertex of one partition is the center of a polytope of the other partition. The Voronoi partition consists of Voronoi polytopes with its centers in lattice points. Moreover, any polytope of the Voronoi partition is obtained by a translation of the Voronoi polytope with the center in the origin (if it is a lattice point). Below we suppose that origin is the zero point of L and call this polytope the Voronoi polytope. It consists of those points of R that are at least as closed to 0 as to any other lattice point. The Delaunay partition consists of Delaunay polytopes which are, in general, not congruent. The set of all Delaunay polytopes having 0 as a vertex is called the star of Delaunay polytopes.