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 Magazinov, Alexander, "VORONOI CONJECTURE FOR FIVE-DIMENSIONAL PARALLELOHEDRA" (2019). Mathematical and Statistical Sciences Faculty Publications and Presentations. 85.