Gruber P. Convex and Discrete Geometry

84 Convex Bodies
similarly, D j is a subset of a translate of Ci, say. This is possible only if C1 = Ci.
Hence C1 = D j − s j. Continuing in this way, we obtain m = n and {C1,...,Cm} =
{D1 − s1,...,Dm − sm}. Hence
D1 ⊕···⊕ Dm − (s1 +···+sm) = C1 ⊕···⊕Cm = D1 ⊕···⊕ Dm.
This can hold only if s j = 0 for all j. Hence {C1,...,Cm} = {D1,...,Dm},
concluding the proof of the uniqueness. ⊓⊔
Hausdorff Metric and the Natural Topology on C
The space C of convex bodies in E d is endowed with a natural topology. It can be introduced
as the topology induced by the Hausdorff metric δ H on C, which is defined
as follows:
δ H (C, D) = max � max
x∈C min �x − y�, max
y∈D y∈D min
x∈C �x − y�� for C, D ∈ C.
The metric δ H was first defined by Hausdorff [481] in a more general context. A nonsymmetric
version of it was considered earlier by Pompeiu [812] and Blaschke [124]
was the first to put it to use in convex geometry in his selection theorem, see below.
The Hausdorff metric can be defined in different ways.
Proposition 6.3. Let C, D ∈ C. Then:
(i) δ H (C, D) = inf � δ ≥ 0 : C ⊆ D + δB d , D ⊆ C + δB d�
(ii) δ H (C, D) = max � |hC(u) − h D(u)| :u ∈ S d−1�
(iii) δ H (C, D) = maximum distance which a point of one of the bodies C, D
can have from the other body
Proof. Left to the reader. ⊓⊔
If we consider a topology on C or on a subspace of it, such as Cp, it is always assumed
that it is the topology induced by δ H .
The Blaschke Selection Theorem
In many areas of mathematics there is need for results which guarantee that certain
problems, in particular extremum problems, have solutions. Examples of such results
are the Bolzano–Weierstrass theorem for sequences in R, the Arzelà–Ascoli theorem
for uniformly bounded equicontinuous families of functions and the selection Theorem
25.1 of Mahler for (point) lattices.
In convex geometry, the basic pertinent result is Blaschke’s selection theorem
[124] for convex bodies. It can be used to show that, for example, the isoperimetric
problem for convex bodies has a solution. Here we give the following version of it,