Gruber P. Convex and Discrete Geometry

32 Tiling with Convex Polytopes 479
This conjecture has been proved for d = 2, 3, 4 by Delone [256] and Engel
[298] established for d = 5 the slightly weaker result that each parallelohedron
is combinatorially equivalent to a Dirichlet–Voronoĭ cell of a lattice. For general d
Voronoĭ [1013] proved the conjecture if P is the prototile of a primitive lattice tiling,
that is, the tiling is facet-to-facet and at each vertex of a tile there meet precisely
d + 1 tiles. ˇZitomirskiĭ [1052] observes that Voronoĭ needs in his proof only that
at each sub-facet of a tile there meet precisely three tiles and gives a far-reaching
generalization of Voronoĭ’s result, but his proof is difficult to understand. If P is a
zonotope, then the conjecture is true according to Erdahl [306]. Unfortunately there
are convex polytopes which are lattice tiles but not zonotopes, see the remark on
parallelohedra and zonotopes in the last section. There are several statements which
are equivalent to Voronoĭ’s conjecture, see Deza and Grishukhin [265].
On Hilbert’s 18th Problem
It follows from the earlier results of Venkov–McMullen that a convex polytope which
provides a translative tiling of E d , necessarily is a parallelohedron. Considering this
result, the following conjecture is highly plausible:
Conjecture 32.2. Let P be a convex polytope which provides a tiling of E d by congruent
copies. Then P is a stereohedron that is, there is a crystallographic group G
such that {gP : g ∈ G} is a tiling of E d .
Hilbert seems to have had doubts whether this was true since, in his 18th problem,
he asked the following:
The question arises: whether polyhedra also exist which do not appear as fundamental
regions of groups of motions, by means of which nevertheless by a suitable
juxtaposition of congruent copies a complete filling up to all space is possible.
See [501]. Reinhardt [828] produced the first example of a (non-convex) polytope P
in E 3 which yields a tiling with congruent copies but no tiling by means of the copies
of P under the rigid motions of a crystallographic group. The first (non-convex)
planar examples are due to Heesch [485] and the first convex example in E 2 was
given by Kershner [579]. In all these examples the congruent copies of P are not
just translates. It is thus of particular interest that Szabó [980] exhibited an example
of a (non-convex) polytope in E 3 which tiles by translation but not by means of a
crystallographic group.
The answer to Hilbert’s problem is in the negative and thus the Conjecture false.
Hence the following problem arises.
Problem 32.1. Specify mild additional conditions which imply that a convex polytope
in E d which tiles E d with congruent copies, is a stereohedron.
Conjectures of Minkowski and Keller
Minkowski [735] conjectured that, in any lattice tiling of a cube, there are translates
of this cube which meet facet-to-facet and proved this for d = 2, 3. The general