Gruber P. Convex and Discrete Geometry

118 Convex Bodies
Since, by Volland’s polytope extension Theorem 7.2, φ satisfies, on P, the
inclusion–exclusion formula, it follows that
φ(Q1n ∪···∪ Qmn) = �
I
(−1) |I |−1 φ(Q In).
Then, letting n →∞, the continuity of φ on C together with (17)–(19) implies that
φ(C1 ∪···∪Cm) = �
I
(−1) |I |−1 φ(CI ).
Thus the inclusion–exclusion formula is valid for φ on C, concluding the proof of the
theorem. ⊓⊔
The Euler Characteristic on L(C)
The Euler characteristic on L(C) is a valuation χ which is defined as follows:
First, let
χ(C) = 1forC ∈ C and χ(∅) = 0.
This, clearly, is a continuous valuation on C. By Groemer’s extension theorem it
extends uniquely to a valuation χ on L(C), theEuler characteristic on L(C). Since
L(C) is a lattice, χ satisfies the inclusion–exclusion formula. Thus:
(20) χ(C1 ∪···∪Cm) = m − # � (i1, i2) : i1 < i2, Ci1 ∩ Ci2 �=∅�
+ # � (i1, i2, i3) : i1 < i2 < i3, Ci1 ∩ Ci2 ∩ Ci3 �=∅� −···
for C1,...,Cm ∈ C,
where # is the counting function.
7.2 Elementary Volume and Jordan Measure
Volume is one of the seminal concepts in convex geometry, where its theory is now
part of the theory of valuations.
In this section, we define the notions of elementary volume of boxes and of volume
or Jordan measure and establish some of their properties. In particular, it will
be shown that both are valuations. Jordan measure plays an important role in several
other sections. In some cases we use their properties not proved in the following,
for example, Fubini’s theorem or the substitution rule for multiple integrals. We also
make some remarks about the problems of Busemann-Petty and Shephard and the
slicing problem.
In Section 16.1, the elementary volume on the space of convex polytopes will be
treated in a similar way.