Proof of the Voronoi conjecture for 3-irreducible parallelotopes

This article proves the Voronoi conjecture on parallelotopes in the special case of 3-irreducible tilings. Parallelotopes are convex polytopes which tile the Euclidean space by their translated copies, like in the honeycomb arrangement of hexagons in the plane. An important example of parallelotope is the Dirichlet-Voronoi domain for a translation lattice. For each point $x$ in a translation lattice, we define its Dirichlet-Voronoi (DV) domain to be the set of points in the space which are at least as close to $x$ as to any other lattice point. The Voronoi conjecture, formulated by the great Ukrainian mathematician George Voronoi in 1908, states that any parallelotope is affinely equivalent to the DV-domain for some lattice. Our work proves the Voronoi conjecture for 3-irreducible parallelotope tilings of arbitrary dimension: we define the 3-irreducibility as the property of having only irreducible dual 3-cells. This result generalizes a theorem of Zhitomirskii (1927), and suggests a way to solve the conjecture in the general case.