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 $E_6$.
Keywords:point lattice, superposition of lattice layers, contact vectors.
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