Gruber P. Convex and Discrete Geometry

32 Tiling with Convex Polytopes 471
This sub-section contains a proof of a characterization of translative convex
tiles due to Venkov [1008], Alexandrov [17] and, independently but much later,
McMullen [710]. The proof is difficult, a slightly different version of the proof is
due to Zong [1048] and in [712] McMullen has given an outline. The Venkov–
Alexandrov–McMullen theorem shows that, in particular, a translative convex tile
necessarily is a lattice tile or parallelohedron. Thus it is an example of the phenomenon
that an object which has a certain property, in some cases has an even stronger
such property, see the heuristic remark in Sect. 2.1.
For extensions and auxilliary results we refer to Alexandrov [17] and Groemer
[400]. For remarks on earlier results consult [710, 711].
A conjecture of Voronoĭ asserts, that parallelohedra are affine images of
Dirichlet–Voronoi cells of lattices. For the conjecture and some pertinent results see
Sect. 32.3. For Dirichlet–Voronoĭ cells Ryshkov and Bol’shakova [869] proved an
interesting decomposition theorem.
Necessity of the Conditions
First, some notation is introduced. Let P be a (proper) centrally symmetric convex
polytope in E d , each facet of which is also centrally symmetric. A sub-facet,orridge
of P is a (d − 2)-dimensional face of P. Given a sub-facet G of P, thebelt of P
corresponding to it consists of all facets of P which contain translates of G and −G
as faces.
Theorem 32.2. Let P be a proper convex body in E d which is a translative tile. Then
the following claims hold:
(i) P is a convex polytope.
(ii) P is centrally symmetric.
(iii) Each facet of P is centrally symmetric.
(iv) Each belt of P consists of 4 or 6 facets.
Proof. Let {P + t : t ∈ T } be a translative tiling of E d with o ∈ T .
(i) Let s, t ∈ T, s �= t. Then int(P + s) ∩ int(P + t) = ∅. Thus s − t �∈
int P − int P = int(P − P). Since o ∈ int(P − P), it follows that �s − t� is
bounded below by a positive constant. Hence T is a discrete set in E d . This, and the
compactness of P, show that only finitely many translates touch P, say the translates
P +ti, ti ∈ T \{o}, i = 1,...,k, and all other translates P +t, t ∈ T \{t1,...,tk},
have distance from P bounded below by a positive constant. Hence each point of
bd P is contained in one of the finitely many touching sets P∩(P+ti). The separation
theorem 4.4 for convex bodies then shows that each point in bd P is contained in one
of finitely many supporting hyperplanes. This finally implies that the convex body P
is actually a convex polytope.
(ii) Clearly, {−P − t : −t ∈ −T } is also a tiling of E d . Since T and thus
−T are discrete as shown earlier, and since −P is compact, there are only finitely
many translates −P − ti, ti ∈ T, i = 1,...,k, say, which meet P. Thus P is the
union of the k non-overlapping centrally symmetric convex polytopes P ∩(−P −ti),