Geodesic
A Class of Affinely Equivalent Voronoi Parallelohedra
Matematičeskie zametki, Tome 95 (2014) no. 5, pp. 697-707

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Given any parallelohedron $P$, its affine class $\mathscr A(P)$, i.e., the set of all parallelohedra affinely equivalent to it, is considered. Does this affine class contain at least one Voronoi parallelohedron, i.e., a parallelohedron which is a Dirichlet domain for some lattice? This question, more commonly known as Voronoi's conjecture, has remained unanswered for more than a hundred years. It is shown that, in the case where the subset of Voronoi parallelohedra in $\mathscr A(P)$ is nonempty, this subset is an orbifold, and its dimension (as a real manifold with singularities) is completely determined by its combinatorial type; namely, it is equal to the number of connected components of the so-called Venkov subgraph of the given parallelohedron. Nevertheless, the structure of this orbifold depends not only on the combinatorial properties of the parallelohedron but also on its affine properties.
Mots-clés : parallelohedron, Voronoi parallelohedron
Keywords: affinely equivalent parallelohedra, Venkov graph, Venkov subgraph, orbifold of Voronoi parallelohedra. 
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     author = {A. A. Gavrilyuk},
     title = {A {Class} of {Affinely} {Equivalent} {Voronoi} {Parallelohedra}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {697--707},
     publisher = {mathdoc},
     volume = {95},
     number = {5},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2014_95_5_a5/}
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A. A. Gavrilyuk. A Class of Affinely Equivalent Voronoi Parallelohedra. Matematičeskie zametki, Tome 95 (2014) no. 5, pp. 697-707. http://geodesic.mathdoc.fr/item/MZM_2014_95_5_a5/