Perfect Delaunay Polytopes and Perfect Inhomogeneous Forms

A lattice Delaunay polytope D is called perfect if it has the property that there is a unique circumscribing ellipsoid with interior free of lattice points, and with the surface containing only those lattice points that are the vertices of D. An inhomogeneous quadratic form is called perfect if it is determined by such a circumscribing ''empty ellipsoid'' uniquely up to a scale factor. Perfect inhomogeneous forms are associated with perfect Delaunay polytopes in much the way that perfect homogeneous forms are associated with perfect point lattices. We have been able to construct some 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 are intimately related to the theory of Delaunay polytopes, and to Voronoi's theory of lattice types.