# zbMATH — the first resource for mathematics

Zonotopes, dicings, and Voronoi’s conjecture on parallelohedra. (English) Zbl 0938.52016
The author settles a special case of Voronoi’s conjecture on parallelohedra, by proving the following theorem: A zonotope which admits a facet-to-facet tiling of Euclidean $$d$$-space by translates is affinely equivalent to the Voronoi polytope of a suitable lattice. He also proves that the Voronoi polytope of a lattice is a zonotope if and only if the corresponding Delaunay tiling is a dicing. The proofs make use of B. A. Venkov’s [Vestnik Leningrad. Univ., Ser. Math. Fiz. Him. 9, 11-31 (1954); see also Usp. Mat. Nauk 9, No. 4(62), 250-251 (1954; Zbl 0056.14103)] and P. McMullen’s [Mathematika 27, 113-121 (1980; Zbl 0432.52016)] characterization of parallelohedra.

##### MSC:
 52C22 Tilings in $$n$$ dimensions (aspects of discrete geometry) 52C07 Lattices and convex bodies in $$n$$ dimensions (aspects of discrete geometry) 11H31 Lattice packing and covering (number-theoretic aspects)
Full Text: