We prove the Voronoi conjecture for five-dimensional parallelohedra. Namely, we show that if a convex five-dimensional polytope P tiles R5 with translations, then P is an affine image of the Dirichlet-Voronoi cell for a five-dimensional lattice.
Garber, Alexey, and Alexander Magazinov. 2020. “Voronoi Conjecture for Five-Dimensional Parallelohedra.” ArXiv:1906.05193 [Math], June. http://arxiv.org/abs/1906.05193.