The Perfect Lattices Γ(Un), and the Covering Density of Γ(U9)

A description of perfect lattices Γ(Un) generated by the L-polytopes Un is given. Γ(Un) is the Coxeter lattice Arn for r = (n + 1)/2. It is proved that the covering density of Γ(U9) is less than that of A*9 , the dual of the root lattice A9 . It is shown that all the L-polytopes of Γ(U7) are congruent to U7.