Gruber P. Convex and Discrete Geometry

236 Convex Bodies
Hence ν = 0 and thus s + q = t and it follows that
3n =�q� =�t − s� ≤�t�+�s� ≤2n.
This contradiction concludes the proof of (3). Now, noticing that the sets Cn +[o, p],
p ∈ 3nS d−1 , all are isometric to Cn, the proof of (2) is complete.
A simple, well-known measure-theoretic result says that a measure cannot be
σ -finite if there is an uncountable family of pairwise disjoint sets with positive measure.
This together with the isometry-invariance of µ and statements (1) and (2)
implies that µ is not σ -finite, which yields the desired contradiction. ⊓⊔
13.3 On the Metric Structure of C
There is a long list of metrics and other notions of distance on C and certain subspaces
of it, including Cp, that have been studied. See, e.g. [428,429]. The most import ones
are the Hausdorff metric δ H on C, the symmetric difference metric δ V on Cp, and the
Banach–Mazur distance on the space of all (equivalence classes with respect to nonsingular
linear transformations of) proper, o-symmetric convex bodies. While a good
deal of all articles in convex geometry make use of such metrics, the metric spaces
〈C,δ H 〉, etc. per se have rarely been investigated. Of what is known, we mention
estimates for ε-nets of � C(B d ), δ H � , � Cp(B d ), δ V � and characterizations of isometries.
In the following two characterizations of isometries will be stated. The proofs
are rather long and technical and thus are omitted. For related results, see the author
[428].
Description of the Isometries of 〈C,δ H 〉
Refining earlier work of Schneider [902], Gruber and Lettl [449] showed the following
result.
Theorem 13.4. Let I : C → C be a mapping. Then the following statements are
equivalent:
(i) I is an isometry with respect to δ H .
(ii) There are a rigid motion m : E d → E d and a convex body D, such that
I (C) = mC+ DforC∈ C.
Remark. The Hausdorff metric δ H clearly may be extended to the space K = K(E d )
of all compact subsets of E d . Generalizing the above result, Gruber and Lettl [448]
characterized the isometries of 〈K,δ H 〉.
Description of the Isometries of 〈C p,δ V 〉
A result of the author [415] is as follows.
Theorem 13.5. Let I : Cp → Cp be a mapping. Then the following statements are
equivalent:
(i) I is an isometry with respect to δ V .
(ii) There is a volume-preserving affinity a : E d → E d such that
I (C) = aC forC ∈ Cp.