Gruber P. Convex and Discrete Geometry

472 Geometry of Numbers
i = 1,...,k. An application of Minkowski’s symmetry theorem 18.3 then shows that
P itself is centrally symmetric.
A different proof of (ii) can be obtained along the following lines: the tiling
{P + t : t ∈ T } clearly has density 1. This density is V (P)δ(T ) where δ(T ) is the
density of the discrete set T . {P + t : t ∈ T } is a tiling and thus a packing. Then
� 12 (P − P) + t : t ∈ T � is also a packing by Proposition 30.4. It thus has density
V � 1 2 (P − P) � δ(T ) ≤ 1. To prove (ii), assume that P is not centrally symmetric.
Then P and −P are not homothetic and the Brunn–Minkowski theorem 8.1 yields
the contradiction:
� �
1
1 ≥ V (P − P) δ(T )>V (P)δ(T ) = 1.
2
(iii) By (ii) we may assume that o is the centre of P. LetF be a facet of P and
−F its opposite facet. Each point in the relative interior of F is contained in the facet
−F + t of a suitable translate P + t, t ∈ T \{o}, of the given tiling (Fig. 32.3). This
implies that the relative interior of F, and thus also F, is the union of finitely many
non-overlapping, centrally symmetric (d − 1)-dimensional convex polytopes of the
form F ∩ (−F + t). Now apply Minkowski’s symmetry theorem 18.3 (Fig. 32.4).
(iv) Note that we have already proved statements (i)–(iii). Let G be a sub-facet of
P and consider the corresponding belt. It consists of, say, k pairs of opposite facets.
For the proof of (iv) assume that, on the contrary, k ≥ 4. Consider the orthogonal
Fig. 32.3. Lattice tiles in E 2
Fig. 32.4. Lattice tiles in E 3