Gruber P. Convex and Discrete Geometry

88 Convex Bodies
The definition of δnk shows that
� x ∈ B : δnk (x) ≤ ε� ⊆ Cnk + εBd .
Since δnk (x) ≤ δC(x) + ε for all sufficiently large k,
(15) C = � x ∈ B : δC(x) = 0 � ⊆ � x ∈ B : δnk (x) ≤ ε� ⊆ Cnk + εBd
for all sufficiently large k.
ε>0 was chosen arbitrarily. Thus (14) and (15) together with Proposition 6.3 (i)
imply (13).
Propositions (12) and (13) yield (8), concluding the proof of the theorem. ⊓⊔
Other Versions and Generalizations of the Selection Theorem
Sometimes the following versions of Blaschke’s selection theorem are needed in the
context of convex geometry. These are immediate consequences of the above form of
the selection theorem. Here C is endowed with its natural topology and its subspace
Cp of proper convex bodies with the corresponding induced topology.
Theorem 6.4. The following claims hold:
(i) C is a locally compact space.
(ii) Cp is a locally compact space.
(iii) C, endowed with the metric δ H , is a boundedly compact complete metric
space.
Clearly, the Hausdorff metric can be defined for the space of compact sets in any
metric space. The above proofs can easily be generalized to show that the Blaschke
selection theorem holds for spaces of compact sets in boundedly compact metric
spaces endowed with the Hausdorff metric, and even in more general situations.
Such spaces of compact sets are sometimes called hyperspaces. An analogue of the
selection theorem for hyperspaces was first proved by Vietoris [1010]. Related, more
general results are due to Michael [722]. For hyperspaces, see Beer [88] and Illanes
and Nadler [535].
Equivalence of Blaschke’s Selection Theorem and the Arzelà–Ascoli Theorem
Using a general version of Blaschke’s selection theorem, Bol [140] proved the classical
Arzelà–Ascoli theorem. Conversely, Heil [487] proved a general version of
Blaschke’s selection theorem by means of the Arzelà–Ascoli theorem.
6.2 Minkowski’s Theorem on Mixed Volumes and Steiner’s Formula
A highlight of the early Brunn–Minkowski theory is Minkowski’s [744] theorem on
mixed volumes. It says that the volume of a linear combination of convex bodies is