  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.
