  A lattice Delaunay polytope is known as perfect if the only ellipsoid, that
can be circumscribed about it, is its Delaunay sphere. Perfect Delaunay
polytopes are in one-to-one correspondence with arithmetic equivalence classes
of positive quadratic functions on the n-dimensional integral lattice that can
be recovered, up to a scale factor, from the representations of its minimum. We
develop a structural theory of such polytopes and describe all known perfect
Delaunay polytopes in dimensions one through eight. We suspect that this list
is complete.
