Gruber P. Convex and Discrete Geometry

32 Tiling with Convex Polytopes 477
Let u ∈{1,...,n} be the smallest index such that r ∈ Ru and v ∈{1,...,n} the
largest index such that s ∈ Rv. Clearly, u �= v. Since L(li, F) is finite, we can find
two sequences (rw) in (int Ru)∩ N and (sw) in (int Rv)∩ N with rw → r and sw → r
as w →∞, such that the arcs �rwsw meet no face of dimension at most d−k−3ofany
polytope (P+g)∩S, g ∈ L(li, F), (and thus meet no face of dimension at most d−2
of any polytope P + g, g ∈ L(li, F),) and such that the arcs �rwsw pass through the
same sequence S1 = Ru, S2,...,Sa = Rv where S j = (P +m j)∩ S, m j ∈ L(li, F)
for j = 1,...,a. Then, since dim F ≥ k + 1, the inductive assumption shows
that (P + m j) ∩ (P + m j+1) is a common facet of both P + m j and P + m j+1
containing F and G for j = 1,...,a − 1. Finally, since �rwsw → �rs, we see that
P + l = P + l1,...,P + lu, P + m1,...,P + ma, P + lv,...,P + ln = P + m is
anew{l, m}-chain with which the loop E is associated. Since |E| < |D| =λ,thisis
impossible. The proof of the induction and thus of (10) is complete.
To conclude the proof of (9) we now show that
(11) int(P + l) ∩ int(P + m) =∅for l, m ∈ L, l �= m.
It follows, from the second step, that there is a number δ > 0 such that the
δ-neighbourhood of P + l, l ∈ L, is contained in the set
� � � � � �
P + m : m ∈ L(l, G : G face of P + l .
Let i be the number of translates P +m in this set. Clearly, this set is contained in the
union of all sequences of translates of P starting with P +l which can be obtained by
fitting together facet-to-facet at most i translates of P. This union obviously contains
the δ-neighbourhood of P + l. Applying the same process to each translate of this
first union, we obtain a second union of translates of P, which contains the 2δneighbourhood
of P + l, etc. Continuing in this way, we arrive at the following
statement:
(12) Let λ>0 and l ∈ L. Then there is a number j such that the union of
all sequences of translates of P starting with P + l which can be obtained
by fitting together facet-to-facet at most j translates of P contains the 2λneighbourhood
of P + l.
To prove (11), assume the contrary. Then there are l, m ∈ L, l �= m, such that
int(P + l) ∩ int(P + m) �= ∅. Call a sequence P + l = P + l1, P + l2,...,P + ln =
P+m, li ∈ L, li �= l j, for i �= j,an{l, m}-chain if (P+li)∩(P+li+1) is a common
facet of both P +li and P +li+1 for i = 1,...,n−1. Choose p ∈ int(P +l)∩int(P +
m). Define a loop based at p associated with the {l, m}-chain P + l1,...,P + ln
to be a closed polygon C starting at p which can be dissected into line segments
C1 ⊆ P + l1,...,Cn ⊆ P + ln, such that Ci and Ci+1 meet in (P + li) ∩ (P + li+1)
for i = 1,...,n and ln+1 = l1 . C is an interior loop if each line segment Ci is
contained in relint � (P +li−1)∩(P +li) � ∪int(P +li)∪relint � (P +li)∩(P +li+1) �
for i = 1,...,n and ln+1 = L1.Letλ be the infimum of the lengths of interior loops.
Consider all loops of length at most 2λ. Then using (8) and a similar compactness
argument to that in the third step, we see that there is a loop of length λ. Now arguing