Gruber P. Convex and Discrete Geometry

478 Geometry of Numbers
as in the third step we arrive at a contradiction, concluding the proof of (11) and thus
of (9).
Finally, the definition of L in (4) together with Proposition (9) implies by
Theorem 21.2 that L is a lattice. Then (6) and (9) show that L provides a tiling
of E d with prototile P. ⊓⊔
Parallelohedra and Zonotopes
A zonotope is a finite sum of line segments. It thus has the property that its faces
of all dimensions are centrally symmetric. If, conversely, a convex polytope has the
property that for k = 2 in case d = 2, 3 and for a given k ∈{2,...,d − 2} in case
d ≥ 4, its k-dimensional faces are all centrally symmetric, then it is already a zonotope
as shown by McMullen [706]. Hence, all 2− and 3-dimensional parallelohedra
are zonotopes. The example of the so-called 24-cell in E 4 , which is a parallelohedron,
shows that a parallelohedron may have faces which are not centrally symmetric and
thus is not a zonotope. For information on the 24-cell, see Coxeter [230].
Non-Convex Tilings are Different
The convexity assumption in Theorem 32.3 cannot be omitted. Stein [956] specified
simple star bodies in E 5 and E 10 which tile by translations but do not admit lattice
tilings. See Stein and Szabó [957].
32.3 Conjectures and Problems of Voronoĭ, Hilbert,
Minkowski, and Keller
There is a small series of tiling problems with marked impact on the literature in discrete
geometry and convexity in the twentieth century. We mention the conjecture of
Voronoĭ, Hilbert’s 18th problem, the cube conjecture of Minkowski and its stronger
version due to Keller.
In this section these problems are described and some references to the literature
given.
Conjecture of Voronoĭ
A characterization of convex lattice tiles different from the earlier characterization
of Venkov, Alexandrov and McMullen, is indicated by the following conjecture of
Voronoĭ on parallelohedra.
Conjecture 32.1. Let P be a proper convex polytope which admits a lattice tiling of
E d , i.e. P is a parallelohedron. Then there is a lattice L such that P is a suitable
linear image of the Dirichlet–Voronoĭ cell
� x :�x� ≤�x − l� for all l ∈ L � .