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 ${\text{sfA}}_9$ so that the resulting perfect lattice, called ${\text{sfA}}_9^2$ 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 $s\ge 2$ and $n+1\ge 4s$ 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 ${\text{sfA}}_9$ and ${\text{sfA}}_9^2$.