Perfect but not generating Delaunay polytopes

In his seminal 1951 paper "Extreme forms" Coxeter \cite{cox51} observed that for \(n \ge 9\) one can add vectors to the perfect lattice \(\sfA_9\) so that the resulting perfect lattice, called \(\sfA_9^2\) by Coxeter, has exactly the same set of minimal vectors. An inhomogeneous analog of the notion of perfect lattice is that of a lattice with a perfect Delaunay polytope: the vertices of a perfect Delaunay polytope are the analogs of minimal vectors in a perfect lattice. We find a new infinite series \(P(n,s)\) for \(s\geq 2\) and \(n+1\geq 4s\) of \(n\)-dimensional perfect Delaunay polytopes. A remarkable property of this series is that for certain values of \(s\) and all \(n \ge 13\) one can add points to the integer affine span of \(P(n,s)\) in such a way that \(P(n,s)\) remains a perfect Delaunay polytope in the new lattice. Thus, we have constructed an inhomogeneous analog of the remarkable relationship between \(\sfA_9\) and \(\sfA_9^2\).