How to compute the rank of a Delaunay polytope

Roughly speaking, the rank of a Delaunay polytope is its number of degrees of freedom. In [M. Deza, M. Laurent, Geometry of Cuts and Metrics, Springer Verlag, Berlin, Heidelberg, 1997], a method for computing the rank of a Delaunay polytope PP, using the hypermetrics related to PP, is given. Here a simpler more efficient method, which uses affine dependencies instead of hypermetrics, is given. This method is applied to the classical Delaunay polytopes: cross-polytopes and half-cubes.Then, we give an example of a Delaunay polytope, which does not have any affine basis.