Gruber P. Convex and Discrete Geometry

16 Volume of Polytopes and Hilbert’s Third Problem 287
F ′
ij = G′ ij = F′ i ∩ G′ j for i = 1,...,m, j = 1,...,n.
(Some pieces may be empty.) Now add to P those cylinders of the form Fij +[o, t]
which are proper. This gives a dissection of P +[o, t]. Next remove from P +[o, t]
those cylinders of the form Gij +[o, t] which are proper. Then P + t remains. Since
by (13) V (Fij +[o, t]) = V (Gij +[o, t]), Proposition (10) shows that V (P) =
V (P + t). The induction for (14) is thus complete.
To prove (15), let P ∈ P and choose p ∈ P. Then o ∈ P − p. By induction,
v ≥ 0. The definition of V in (9) then yields V (P − p) ≥ 0 and thus V (P) =
V (P − p) ≥ 0 by (14), concluding the induction for (15).
For the proof of (16), let P, Q ∈ P with P ⊆ Q. IfP is improper, V (P) = 0 ≤
V (Q) by (13) and (15). Assume now that P is proper. The hyperplanes through the
facets of P dissect Q into polytopes P, Q1,...,Qm ∈ Pp, say. (15) and (10) then
imply that
V (P) ≤ V (P) + V (Q1) +···+V (Qm) = V (P ˙∪Q1 ˙∪··· ˙∪Qm) = V (Q)
and the induction for (16) is complete.
Having proved (10)–(16) for d, the induction is complete. Thus (10)–(16) and, in
particular, the theorem hold generally. ⊓⊔
Simple Consequences
An important property of V is the following.
Corollary 16.1. V is rigid motion invariant.
Proof. Let r : E d → E d be a rigid motion. Define a mapping W : P → R
by W (P) = V (rP) for P ∈ P. Since V (·) is a simple, translation invariant
monotone valuation on P by Theorem 16.2, it is immediate that this also holds
for W (·). The proof of statement (11) in the proof of Theorem 7.5 then yields
W ([0, 1] d ) = V ([0, 1] d ) = 1. An application of the uniqueness theorem 16.1 now
shows that V = W ,orV (P) = V (rP) for P ∈ P. ⊓⊔
The definition of V (·) and an easy induction argument yields the following
property.
Proposition 16.1. V is positively homogeneous of degree d.
Volume and Elementary Volume
By Theorems 7.5 and 16.2 the volume (or Jordan measure) and the elementary volume
are both simple, monotone and translation invariant valuations on P and their
values for the unit cube both are 1. Thus Theorem 16.1 yields the following result:
Corollary 16.2. On P volume, i.e. Jordan measure, and elementary volume coincide.
Similarly, Minkowski surface area and elementary surface area, i.e. the sum of the
elementary volumes of the facets, coincide on P.