On Voronoi’s conjecture for four- and five-dimensional parallelohedra

1. Parallelohedra and Voronoi’s conjecture. A convex d-dimensional polytope is called a parallelohedron or a d-parallelohedron if there is a tiling of the space R into parallel copies of P . In particular, all parallelograms and all hexagons with a centre of symmetry are 2-parallelohedra. All five types of 3-parallelohedra were classified by Fedorov at the end of the 19th century. The theory of parallelohedra has its origins in the works by Fedorov, Minkowski, Voronoi, and Delone. Parallelohedra are closely connected with mathematical crystallography, the classification of crystallographic groups, algorithmic and geometric questions relating to integer lattices, and, in particular, with Hilbert’s 18th problem. Voronoi’s conjecture [1] is one of the central conjectures in the theory of parallelohedra. It states that for each d-parallelohedron P , there is a d-dimensional lattice Λ such that P is affinely equivalent to the Dirichlet–Voronoi cell of Λ. If Voronoi’s conjecture holds for P , then we call it a V-parallelohedron. Voronoi’s conjecture has been proved fully for d ⩽ 5. The cases d = 1, 2, 3 are common wisdom. The proof for d = 4 was given by Delone [2] in 1929. For d = 5, the proof was obtained by the authors of the present paper in 2019; see [3]. A review of key results in the theory of parallelohedra can be found in [4], Chap. 3, and in [5]. In this note we present a new proof of Voronoi’s conjecture in R, which uses ideas from [3] adapted for d = 4. For instance, our proof relies on a combinatorial approach, in contrast to Delone’s geometric methods. Both approaches use a number of general properties of parallelohedra, and, in particular, a classification of the types of coincidence of parallelohedra at faces of codimension three and the existence of a layered structure of tilings into parallelohedra under certain constraints. However, we rely on combinatorial methods developed long after Delone’s publication. In conclusion, we present a sketch of the proof of Voronoi’s conjecture for d = 5 from [3].