Voronoi-Dickson Hypothesis on Perfect Forms and L-types

Abstract

George Voronoi (1908, 1909) introduced two important reduction methods for positive quadratic forms: the reduction with perfect forms, and the reduction with L-type domains, often called domains of Delaunay type. The first method is important in studies of dense lattice packings of spheres. The second method provides the key tools for finding the least dense lattice coverings with equal spheres in lower dimensions. In his investigations Voronoi heavily relied on that in dimensions less than 6 the partition of the cone of positive quadratic forms into L-types refines the partition of this cone into perfect domains. Voronoi conjectured implicitely and Dickson (1972) explicitely that the L-partition is always a refinement of the partition into perfect domains. This was proved for n =< 5 (Voronoi, Delaunay, Ryshkov, Baranovskii). We show that Voronoi-Dickson conjecture fails already in dimension 6.