Gruber P. Convex and Discrete Geometry

6 Mixed Volumes and Quermassintegrals 85
where a sequence of convex bodies is bounded, if it is contained in a suitable (solid
Euclidean) ball.
It turns out that the selection theorems of Arzelà–Ascoli, Blaschke and Mahler
are closely connected, see the remarks at the end of this section and Groemer’s proof
of Mahler’s theorem outlined in Sect. 25.2.
Theorem 6.3. Any bounded sequence of convex bodies in E d contains a convergent
subsequence.
We present two proofs. The first proof, in essence, is due to Blaschke, the second one
to Heil [487].
Proof (using ε-nets). Let C1, C2, ···∈C be contained in a ball B. For the proof that
the sequence C1, C2,..., contains a convergent subsequence, the following will be
shown first:
(1) The sequence C1, C2,...,contains a subsequence D1, D2,...,such that
δ H (Dm, Dn) ≤
For the proof of (1) the main step is to prove that
1
for m, n = 1, 2,...
2min {m,n}
(2) There are sequences C11, C12,...; C21, C22,...; ...,
where C11, C12,... is a subsequence of C1, C2,...,and each subsequent
sequence is a subsequence of the sequence preceding it, such that
δ H (Cmi, Cmj) ≤ 1
for m = 1, 2,..., and i, j = 1, 2,...
2m The first step of the induction is similar to the step from m to m+1, thus only the latter
will be given. Let m ≥ 1 and assume that the first m sequences have been constructed
already and satisfy the inequality for 1,...,m. Since the ball B is compact, it can be
covered by a finite family of balls, each of radius 1/2 m+2 with centre in B. To each
convex body in B we associate all balls of this family which intersect it. Clearly,
these balls cover the convex body. Since there are only finitely many subfamilies
of this family of balls, there must be one which corresponds to each convex body
from an infinite subsequence of Cm1, Cm2,...,say Cm+11, Cm+12,... Now, given
i, j = 1, 2,...,for any x ∈ Cm+1 i there is a ball in our subfamily which contains x.
Hence �x − c� ≤1/2 m+2 , where c is the centre of this ball. This ball also intersects
Cm+1 j. Thus we may choose y ∈ Cm+1 j with �y − c� ≤1/2 m+2 . This shows that,
for each x ∈ Cm+1 i, there is y ∈ Cm+1 j with �x − y� ≤1/2 m+1 . Similarly, for
each y ∈ Cm+1 j there is x ∈ Cm+1 i with �x − y� ≤1/2 m+1 . Thus
δ H � Cm+1 i, Cm+1 j
� 1
≤ for i, j = 1, 2,...
2m+1 by Proposition 6.3(iii). The induction is thus complete, concluding the proof of (2).
By considering the diagonal sequence D1 = C11, D2 = C22,...,we see that (1) is
an immediate consequence of (2).