Gruber P. Convex and Discrete Geometry

Dissections
16 Volume of Polytopes and Hilbert’s Third Problem 281
A polytope P ∈ Pp is dissected into the polytopes P1,...,Pn ∈ Pp, in symbols,
P = P1 ˙∪··· ˙∪Pm,
if P = P1 ∪ ··· ∪ Pm and the polytopes Pi have pairwise disjoint interiors.
{P1,...,Pm} then is said to be a dissection of P.
Uniqueness of Elementary Volume
The elementary volume on the space P of convex polytopes is a valuation with special
properties. We first show that there is at most one such valuation.
Theorem 16.1. Let Φ, Ψ be simple, translation invariant, monotone valuations on
P with Φ([0, 1] d ) = Ψ([0, 1] d ) = 1. Then Φ = Ψ .
Proof (by induction on d). If d = 1, then it is easy to see that Φ([α, β]) =|α −β| =
Ψ([α, β]) for all intervals [α, β] ⊆R, see the corresponding argument for boxes in
the proof of Theorem 7.6.
Assume now that d > 1 and that the theorem holds for dimension d − 1. Since
Φ and Ψ both are valuations on P, they satisfy the inclusion–exclusion principle
by Volland’s extension theorem 7.2. The assumption that Φ and Ψ are simple then
implies that they are simply additive:
(1) Let P = P1 ˙∪··· ˙∪Pm, where P, P1,...,Pm ∈ Pp.
Then Φ(P) = Φ(P1) +···+Φ(Pm) and similarly for Ψ .
Thus the translation invariance of Φ and Ψ together with Φ([0, 1] d ) =
Ψ([0, 1] d ) = 1 shows that
(2) Φ(K ) = Ψ(K ) = 1 for any cube K in E d of edge-length 1,
see the proof of statement (11) in the proof of Theorem 7.5.
Next, the following will be shown.
(3) Φ(Z) = Ψ(Z) for each right cylinder Z ∈ P of height 1.
Let H be a hyperplane and u a normal unit vector of H. Consider the functions
ϕ,ψ : P(H) → R defined by
ϕ(Q) = Φ(Q +[o, u]), ψ(Q) = Ψ(Q +[o, u]) for Q ∈ P(H).
It is easy to see that ϕ and ψ are simple, translation invariant, monotone valuations
on P(H). IfL is a cube of edge-length 1 in H, then L +[o, u] is a cube in E d of
edge-length 1, and thus
ϕ(L) = Φ(L +[o, u]) = 1 = Ψ(L +[o, u]) = ψ(L)