Gruber P. Convex and Discrete Geometry

Define
32 Tiling with Convex Polytopes 475
Q(l, G) = � � � � P + m : m ∈ L(l, F) � : F face of P + l, G � F �
for l ∈ L(o, G).
Let l ∈ L(o, G). Each point of (P + l) ∩ S lies in the relative interior of a suitable
face F of P + l with dim F > dim G = k. The induction assumption thus shows
that each point of (relint F) ∩ S has a neighbourhood in S which is contained in
� � P + m : m ∈ L(l, F) � ⊆ Q(l, G).
Since (P + l) ∩ S is compact, there is a δ>0 such that the δ-neighbourhood of
(P + l) ∩ S in S is contained in Q(l, G). AsL(o, G) is finite by (5), we may take
the same δ for each l ∈ L(o, G). Hence the δ-neighbourhood in S of each point of
the set
R = � � P + l : l ∈ L(o, G) � ∩ S ⊆ S
is contained in
� � Q(l, G) : l ∈ L(o, G) � = � � P + l : l ∈ L(o, G) � .
Since S is arcwise connected and R compact and non-empty, this can hold only if
R = S, concluding the proof of (8). This concludes the induction and thus proves (7).
P is the disjoint union of the sets relint F where F ranges over the 0, 1,...,ddimensional
faces of P. It thus follows from (7) that P ⊆ int � {P + l : l ∈ L}.
Taking into account the fact that P is compact, a suitable δ-neighbourhood of P
is contained in int � {P + l : l ∈ L} too. By periodicity the δ-neighbourhood of
� {P + l : l ∈ L} is also contained in int � {P + l : l ∈ L}. This is possible only if
� {P + l : l ∈ L} =E d . The proof of (6) is complete.
The third step is to show that
(9) {P + l : l ∈ L} is a packing of E d .
In order to show this, we first prove that
(10) for k = d, d − 1,...,0, we have the equality int(P + l) ∩ int(P + m) =∅
for l, m ∈ L(o, F), l �= m, and any face F of P with dim F = k.
Clearly, (10) holds for k = d (where L(o, P) ={o}) and k = d−1 (where L(o, F) =
{o, tF}). Ifk = d − 2, Theorem 32.2 implies that L(o, F) ={o, l, m} with suitable
l, m ∈ L (when the belt corresponding to F consists of six facets) or L(o, F) =
{o, l, m, l + m} (when the belt corresponding to F consists of four facets) and the
translates P + l, l ∈ L(o, F), have pairwise disjoint interiors. Hence (10) holds for
k = d − 2 too. Assume now that k < d − 2 and that (10) holds for all faces F
of P with dim F = k + 1. Let G be a face of P with dim G = k. For the proof
that (10) holds for G, assume the contrary. Then there are l, m ∈ L(o, G), l �= m,
such that int(P + l) ∩ int(P + m) �= ∅. By the definition of L(o, G) we can find
points l = l1, l2,...,ln = m ∈ L(o, G), li �= l j, for i �= j, such that each set