Infinite serie of extreme Delaunay polytopes

A Delaunay polytope $P$ is said to be {\em extreme} if the only (up to isometries) affine bijective transformations $f$ of ${\mathbb{R}}^n$, for which $f(P)$ is again a Delaunay polytope, are the homotheties. This notion was introduced in \cite{DGL92}; also some examples in dimension 1, 6, 7, 15, 16, 22, 23 were constructed and it was proved that in dimension less than 6 there are no extreme Delaunay polytopes, except the segment. In this note, for every $n\geq 6$webuildanextremeDelaunaypolytope$ED_n$ofdimension$n$.