Layer Superposition of the Root Lattice $A_n$

An explicit description of a Dirichlet–Voronoi (DV) cell of the root lattice $A_n$ in the form of the convex hull of an $n$-dimensional parallelotope with a vertex at 0 and its copies centrally symmetric with respect to 0 is given. Remarkable properties of this DV-cell are described, which manifest themselves when layers of the lattice $A_n$ are superposed. It is shown that the one-parameter family of their metric forms runs through the cone of positivity from one boundary to the other, passing through four $L$-type domains.