The six-dimensional Delaunay polytopes

Given a lattice L, a polytope P is called a Delaunay polytope if the set of its vertices is S∩L with S being an empty sphere of the lattice. Extending our previous work (Available from http://il.arXiv.org/abs/math.MG/0108177) on the hypermetric cone HY P7, we classify the six-dimensional Delaunay polytopes according to their combinatorial type. The list of 6241 combinatorial types is obtained by a study of the set of faces of the polyhedral cone HY P7. 11Data and programs are available from http://www.liga.ens.fr/~dutour/DelaunaySix.