Perfect Delaunay Polytopes and Perfect Quadratic Functions on Lattices

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.