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Regular version of the site

Article

The Voronoi conjecture for parallelohedra with simply connected δ-surfaces

Discrete and Computational Geometry. 2015. Vol. 53. No. 2. P. 245-260.
Magazinov A., Gavrilyuk A., Garber A.

We show that the Voronoi conjecture is true for parallelohedra with simply connected δδ-surfaces. That is, we show that if the boundary of parallelohedron PP remains simply connected after removing closed nonprimitive faces of codimension 2, then PP is affinely equivalent to a Dirichlet–Voronoi domain of some lattice. Also, we construct the ππ-surface associated with a parallelohedron and give another condition in terms of a homology group of the constructed surface. Every parallelohedron with a simply connected δδ-surface also satisfies the condition on the homology group of the ππ-surface.