Geodesic
Layer Superposition of the Root Lattice~$A_n$
Matematičeskie zametki, Tome 109 (2021) no. 2, pp. 219-234

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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.
Keywords: Dirichlet–Voronoi cell, root lattice $A_n$, superposition of layers. 
@article{MZM_2021_109_2_a5,
     author = {V. P. Grishukhin},
     title = {Layer {Superposition} of the {Root} {Lattice~}$A_n$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {219--234},
     publisher = {mathdoc},
     volume = {109},
     number = {2},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2021_109_2_a5/}
}
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V. P. Grishukhin. Layer Superposition of the Root Lattice~$A_n$. Matematičeskie zametki, Tome 109 (2021) no. 2, pp. 219-234. http://geodesic.mathdoc.fr/item/MZM_2021_109_2_a5/