Abstract
At the end of his posthumous memoir, Voronoi defined a positive quadratic form, which he denoted by \(\omega(x)\) . Voronoi proved that this form lies on an extreme ray of a simplicial \(L\) -domain adjacent along a facet to the principal \(L\) -domain. In this paper, it is shown that the form \(\omega\) naturally arises as a metric form of the lattices \(L^{n+1}_{Z,D}(h_{n-1})\) , where \(h_{n-1}^2=(n-2)/4\) . These lattices are superpositions of layers of the cubic lattice \(Z^n\) and the root lattice \(D_n\) . In the case of \(D_n\) of odd dimension \(n\) , the lattice \(L^{n+1}_D(h_{n-1})\) has an extremal Delaunay polytope. For \(n=5\) , the lattice \(L^6_D(h_4)\) , where \(h_4^2=3/4\) , is isomorphic to the root lattice \(E_6\) .
