Gruber P. Convex and Discrete Geometry

110 Convex Bodies
A Problem of Santaló
Santaló proposed the following related question.
Problem 6.2. Consider a finite set of geometric functionals F1,...,Fk for convex
bodies in E d , for example the inradius r, the volume V and the diameter diam.
Determine the set �� F1(C),...,Fk(C) � : C ∈ C � ⊆ E k .
As is the case for the above problem of Blaschke, this problem is also difficult in this
generality. For pertinent results see Hernández Cifre, Pastor, Salinas Martínez and
Segura Gomis [497].
7 Valuations
Let S be a family of sets. A (real) valuation on S is a real function φ on S which is
additive in the following sense:
φ(C ∪ D) + φ(C ∩ D) = φ(C) + φ(D) whenever C, D, C ∪ D, C ∩ D ∈ S,
and φ(∅) = 0if∅∈S.
In many cases of interest, S is intersectional, that is, C ∩ D ∈ S for C, D ∈ S.
If S is the space C of convex bodies in E d or a subspace of it such as the space P
of convex polytopes, and in many other cases, the above additivity property is rather
weak since we require additivity only for a small set of pairs C, D in C. In case
where S is the space of Jordan or of Lebesgue measurable sets in E d , the valuation
property coincides with the common notion of additivity in measure theory. Examples
of valuations on C, P and on the space of lattice polytopes, are the volume, the
quermassintegrals, affine surface area, the Dehn invariants in the context of Hilbert’s
third problem and the various lattice point enumerators. Mixed volumes also give
rise to valuations on C.
While special valuations have been investigated since antiquity, it seems that
Blaschke [128], Sects. 41,43, was the first to consider valuations per se and he initiated
their study. Then, his disciple Hadwiger started the systematic investigation
of valuations, culminating in the functional theorems, see [468]. Important later
contributions are due to Groemer, McMullen, Schneider, Betke and Kneser, Klain,
Alesker, Ludwig and Reitzner, and others, see [108, 402, 587, 666, 713, 714, 716].
Amongst others, these contributions deal with Hilbert’s third problem, McMullen’s
polytope algebra, lattice point enumerators and with characterizations and representations
of certain important classes of valuations.
This section contains a small account of the rich theory of valuations. We start
with extension results, introduce the elementary volume and Jordan measure, then
give a characterization of the volume and, as a consequence, show Hadwiger’s functional
theorem. As an application of the functional theorem, the principal kinematic
formula is proved.
For more information on valuations, see Hadwiger [468], McMullen and
Schneider [716], McMullen [714], Klain and Rota [587] and Peri [790]. See also