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≥6n≥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)PDu(n) for all even dimensions n≥6n≥6. An analysis of the first series allows one to construct two two-parametric series P(n;t)P(n;t) and PA(n;t)PA(n;t) of asymmetric extreme Delaunay polytopes such that P(n;1)=PER(n). In addition, for each asymmetric extreme Delaunay polytope PP of dimension nn, an explicit construction of a symmetric extreme Delaunay polytope of dimension n+1n+1 having PP as a section is given.