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This article proves the Voronoi conjecture on parallelotopes in the specialcase of 3-irreducible tilings.  Parallelotopes are convex polytopes which tile the Euclidean space by theirtranslated copies, like in the honeycomb arrangement of hexagons in the plane.An important example of parallelotope is the Dirichlet-Voronoi domain for atranslation lattice. For each point $x$ in a translation lattice, we define itsDirichlet-Voronoi (DV) domain to be the set of points in the space which are atleast as close to $x$ as to any other lattice point. The Voronoi conjecture,formulated by the great Ukrainian mathematician George Voronoi in 1908, statesthat any parallelotope is affinely equivalent to the DV-domain for somelattice.  Our work proves the Voronoi conjecture for 3-irreducible parallelotopetilings of arbitrary dimension: we define the 3-irreducibility as the propertyof having only irreducible dual 3-cells. This result generalizes a theorem ofZhitomirskii (1927), and suggests a way to solve the conjecture in the generalcase.

Proof of the Voronoi conjecture for 3-irreducible parallelotopes