Infinite series of extreme Delaunay polytopes

Recently, Rybnikov and Erdahl [R. Erdahl, K. Rybnikov, Supertopes, Supertopes.pdf at http://faculty.uml.edu/krybnikov/, 2002] constructed an infinite series of asymmetric extreme lattice Delaunay polytopes P ER ( n ) for dimensions n ≥ 6 . Dutour [M. Dutour, An infinite series of extreme Delaunay polytopes, European J. Combin. (2003) (in press)] constructed another infinite series of asymmetric extreme Delaunay polytopes P D u ( n ) for all even dimensions n ≥ 6 . An analysis of the first series allows one to construct two two-parametric series P ( n ; t ) and P A ( n ; t ) of asymmetric extreme Delaunay polytopes such that P ( n ; 1 ) = P ER ( n ) . In addition, for each asymmetric extreme Delaunay polytope P of dimension n , an explicit construction of a symmetric extreme Delaunay polytope of dimension n + 1 having P as a section is given.