Gruber P. Convex and Discrete Geometry

13 The Space of Convex Bodies 231
In the following we first deal with the topology, considering Baire category
results. Then, a result is presented which shows difficulties with the introduction
of measures. Then characterizations of the isometries of the metric spaces 〈C,δ H 〉
and 〈Cp,δ V 〉 are stated without proofs. Finally, we consider the algebraic structure
of C. A characterization of homomorphisms with additional properties of 〈C, +〉 into
〈E d , +〉 is presented. Its proof makes use of spherical harmonics. Results on endomorphisms
of the semigroup 〈C, +〉 and the lattice 〈C, ∧, ∨〉 are stated last.
For more information and additional references, see [428] and the references and
books cited below.
13.1 Baire Categories
A version of Blaschke’s selection theorem says that the spaces C and Cp, endowed
with their natural topologies (which are induced by, e.g. the metric δ H ), are locally
compact, see Theorem 6.4. Thus both are Baire according to a modern form of
Baire’s category theorem. This means that each meagre set has dense complement.
By most or typical convex bodies we mean all convex bodies with a meagre set of
exceptions. For these notions, see Sect. 5.1.
The first Baire category result dealing with spaces of convex bodies is due to Klee
[590]. It says that most convex bodies are smooth and strictly convex. For unclear
reasons it was soon forgotten. Its re-discovery by Gruber [414] some 20 years later
led to a voluminous body of results, see the surveys of Zamfirescu [1039, 1041] and
the author [431]. These results treat:
Differentiability properties
Geodesics
Billiards, normals and mirrors
Approximation
Contact points
Shadow boundaries
Metric projection
Fixed points and attractors
Cut loci and conjugate points
Packing and covering
It is interesting to note that, sometimes, Baire type convexity results are in contrast
to results of differential geometry. For example, a result of the author [423] says
that for most proper convex bodies C in E 3 there is no closed geodesic on bd C
while on each sufficiently smooth proper convex body C in E 3 there are infinitely
many closed geodesics on bd C according to a famous theorem of Bangert [65] and
Hingston [504].
In the following we prove the result of Klee mentioned above. Then an irregularity
criterion will be shown and applied to the approximation of convex bodies.