Gruber P. Convex and Discrete Geometry

i.e.
16 Volume of Polytopes and Hilbert’s Third Problem 283
(8) −Φ(Y1) −···−Φ(Yn) ≤−Ψ(T ) ≤−Φ(Z1) −···−Φ(Zn−1).
Adding (7) and (8) and taking into account that Yi+1 is a translate of Zi, the translation
invariance of Φ implies that
−Φ(Y1) ≤ Φ(T ) − Ψ(T ) ≤ Φ(Y1),
i.e.
�c − b�
|Φ(T ) − Ψ(T )| ≤Φ(Y1) = φ(F) for n ∈ N
n
by (4). Since this holds for all n, it follows that Φ(T ) = Ψ(T ), concluding the proof
of (5). The induction is complete. ⊓⊔
Existence of Elementary Volume
The elementary volume of convex polytopes is to be a simple, translation invariant,
non-negative (equivalently, a monotone) valuation on P, such that for the unit cube
[0, 1] d its value is 1. If it exists, it is unique by what was shown earlier. A candidate
for the elementary volume is the function V : P → R, defined inductively as follows.
For d = 1, let V ([α, β]) =|α− β| for each interval [a, b] ⊆R. Assume now
that d > 1 and that the volume has been defined in dimension d−1. Then it is defined
on each hyperplane and since it is simple, it is 0 on intersections of hyperplanes. This
leads to a notion of elementary volume or area for all (d − 1)-dimensional convex
polytopes in Ed . Denote it by v. Then V is defined by:
(9) V (P) = 1
d
m�
h P(ui)v(Fi) for P ∈ P(Ed ),
i=1
where F1,...,Fm are the facets of P and u1,...,um the corresponding exterior
normal unit vectors of P. IfdimP < d − 1, the definition (9) is to be understood as
V (P) = 0 (empty sum) and in case dim P = d − 1 the polytope P has two facets,
both coinciding with P but with opposite exterior normal unit vectors and (9) yields
V (P) = 0.
The following result shows that V , as defined in (9), has the required properties.
By the uniqueness theorem 16.1 it is the unique such function and thus is legitimately
called the elementary volume on P.
Theorem 16.2. V is a simple, translation invariant, monotone valuation on P with
V ([0, 1] d ) = 1.
Proof. The following assertions will be shown by induction on d:
(10) V is simply additive, that is, V (P) = V (P1) +···+V (Pm)
for P, P1,...,Pm ∈ Pp such that P = P1 ˙∪··· ˙∪Pm.
(11) V is a simple valuation on P.