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 PER(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 PDu(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 PA(n; t) of asymmetric extreme Delaunay polytopes such that P(n; 1) = PER(n). In addition, for each asymmetric extreme Delaunay potytope P of dimension n, an explicit construction of a symmetric extreme Delaunay potytope of dimension n + 1 having P as a section is given.