Gruber P. Convex and Discrete Geometry

476 Geometry of Numbers
(P + li) ∩ (P + li+1) is a common facet of P + li and P + li+1 which contains G as
a face. The sequence of polytopes P +l = P +l1, P +l2,...,P +ln = P + m then
is called an {l, m}-chain. Next, let the small sphere S of dimension d − k − 1 ≥ 2
be chosen as in the proof of (7). Our {l, m}-chain then gives rise to a sequence of
(d − k − 1)-dimensional spherically convex polytopes R1 = (P + l1) ∩ S,...,Rn =
(P + ln) ∩ S on S, called a spherical {l, m}-chain. From the proof of (7), we see
that the spherical polytopes (P + l) ∩ S, l ∈ L(o, G), cover S. By assumption,
int R1 ∩ int Rn = int(P + l) ∩ int(P + m) ∩ S �= ∅.
Choose a point p ∈ int R1 ∩ int Rn such that its antipode in S is not contained
in the spherical hull of any face of any of the spherical polytopes (P + l) ∩ S,
l ∈ L(o, G). Byaloop based at p associated with the spherical {l, m}-chain
R1,...,Rn, we mean a closed spherical polygonal curve C on S starting at p which
can be dissected into the (great circular) arcs C1 ⊆ R1,...,Cn ⊆ Rn such that Ci
and Ci+1 meet in Ri ∩Ri+1. Of course, Cn and C1 meet at p. The loop C is an interior
loop if each arc Ci is contained in relint(Ri−1 ∩ Ri) ∪ int Ri ∪ relint(Ri ∩ Ri+1).
Denote the length of C by |C|. Letλ denote the infimum of the lengths of the interior
loops based at p associated with all possible spherical {l, m}-chains.
To conclude the induction, we will show that both λ = 0 and λ>0 lead to a
contradiction. Clearly, λ = 0 contradicts the definition of loops based at p. It remains
to consider the case λ>0. By (5) and the definition of {l, m}-chains, there are only
finitely many {l, m}-chains. A compactness argument then implies that there is a
loop D based at p with |D| = λ and associated with the spherical {l, m}-chain
R1 = (P + l1) ∩ S,...,Rm = (P + lm) ∩ S, say. An interior loop based at p
associated with a spherical {l, m}-chain can always be deformed into such a loop of
smaller length. Thus D cannot be an interior loop. By our choice of p, most great
circles on S through p avoid all spherical faces of dimension at most d − k − 3of
all spherical polytopes (P + l) ∩ S, l ∈ L(o, G). (To see this project these spherical
faces from p, respectively, its antipode into the equator of S corresponding to the
north pole p.) Each such great circle is an interior loop based at p of length 2π.
Since an interior loop cannot have minimal length, λ