Gruber P. Convex and Discrete Geometry

86 Convex Bodies
For the proof of the theorem it is sufficient to show that
∞� �
(3) D1, D2, ···→ D, where D = Dn + 1
Bd� ∈ C.
2n−1 (1) implies that
and thus,
n=1
D1 + 1
2 Bd ⊇ D2, D2 + 1
2 2 Bd ⊇ D3,...
(4) D1 + B d ⊇ D2 + 1
2 Bd ⊇ ...
Being the intersection of a decreasing sequence of non-empty compact convex sets
(see (3) and (4)), the set D is also non-empty, compact and convex, i.e. D ∈ C. In
order to prove that D1, D2, ···→ D, letε>0. Then
(5) D ⊆ Dn + 1
2n−1 Bd ⊆ Dn + εB d 1
for n ≥ 1 + log2 ε .
Let G = int(D + εB d ). The intersection of the following decreasing sequence of
compact sets
� D1 + B d� \G ⊇
�
D2 + 1
2 Bd�\G
⊇ ...
is contained both in D (see (3)) and in E d \G and thus is empty. This implies that,
from a certain index on, the sets in this sequence are empty. That is,
(6) Dn ⊆ Dn + 1
2 n−1 Bd ⊆ G ⊆ D + εB d for all sufficiently large n.
Since ε>0 was arbitrary, (5) and (6) show that D1, D2, ··· → D, concluding the
proof of (3) and thus of the theorem. ⊓⊔
Proof (using the Arzelà–Ascoli theorem). A special case of the Arzelà–Ascoli
theorem is as follows:
(7) Let B be a ball in E d and f1, f2, ···: B → R a sequence of functions such
that
| fn(x)| ≤const for x ∈ B, n = 1, 2,...,
| fn(x) − fn(y)| ≤�x − y� for x, y ∈ B, n = 1, 2,...
Then the sequence f1, f2,... contains a uniformly convergent subsequence.
For the proof of the selection theorem we have to show the following:
(8) Let B be a ball and C1, C2,... a sequence of convex bodies in B. Then this
sequence contains a convergent subsequence.