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Status:
Available4.7
16 reviewsAbstract
We prove the Voronoi conjecture for five-dimensional parallelohedra. Namely, weshow that if a convex five-dimensional polytope P tiles R5 with translations, thenP is an affine image of the Dirichlet-Voronoi polytope for a five-dimensional lattice.Our proof is based on an exhaustive combinatorial analysis of possible dual 3-cellsand incident dual 4-cells encoding local structures around two-dimensional faces offive-dimensional parallelohedron P and their edges. The analysis is aimed to proveexistence of a free direction for P and is paired with new properties established forparallelohedra (in any dimension) that have a free direction that guarantee the Voronoiconjecture for P.Mathematics Subject Classification 52B20 · 52C07
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