Delaunay and voronoi polytopes of the root lattice E 7 and of the dual lattice E*7

Abstract

We give a detailed geometrically clear description of all faces of the Delaunay and Voronoi polytopes of the root lattice E7 and the dual lattice E*7. Here three uniform polytopes related to the Coxeter-Dynkin diagram of the Lie algebra E7 play a special role. These are the Gosset polytope PGos = 321, which is a Delaunay polytope, the contact polytope 231 (both for the lattice E7), and the Voronoi polytope PV(E*7) = 132 of the dual lattice E*7. This paper can be considered as an illustration of the methods for studying Delaunay and Voronoi polytopes.