  A polytope $D$ whose vertices belong to a lattice of rank $d$ is Delaunay if
there is a circumscribing $d$-dimensional ellipsoid, $E$, with interior free of
lattice points so that the vertices of $D$ lie on $E$. If in addition, the
ellipsoid $E$ is uniquely determined by $D$, we call $D$ perfect. That is, a
perfect Delaunay polytope is a lattice polytope with a circumscribing empty
ellipsoid $E$, where the quadratic surface $\partial E$ both contains the
vertices of $D$ and is determined by them. We have been able to construct
infinite sequences of perfect Delaunay polytopes, one perfect polytope in each
successive dimension starting at some initial dimension; we have been able to
construct an infinite number of such infinite sequences. Perfect Delaunay
polytopes play an important role in the theory of Delaunay polytopes, and in
Voronoi's theory of lattice types.
