Gruber P. Convex and Discrete Geometry

132 Convex Bodies
Seventh,
o
v3
v2
v1
(17) ψ(S) = 0 for each simplex S.
Fig. 7.2. Proof of Proposition (17)
v1 + v2 + v3
Since ψ is simple, (17) holds if dim S ≤ d−1. Assume now that dim S = d and that o
is a vertex of S.Letv1,...,vd denote the other vertices of S and let v = v1+···+vd.
Let P ={λ1v1 +···+λdvd : 0 ≤ λi ≤ 1}. The hyperplanes through v1,...,vd
and v − v1,...,v− vd , respectively, dissect P into S, a centrally symmetric convex
polytope Q, and v − S. (16), (9), (16) and (10) then imply that
0 = ψ(P) = ψ(S) + ψ(Q) + ψ(v − S) = 2ψ(S),
hence ψ(S) = 0, concluding the proof of (17) (see Fig. 7.2).
Finally, since each P ∈ P can be dissected into simplices, (9) and (17) show that
ψ(P) = 0forP ∈ P. The continuity of ψ then implies that
ψ(C) = 0forC ∈ C.
This concludes the induction. The proof of (8) is finished.
To complete the proof of the theorem, let ψ : C → R be defined by ψ(C) =
φ(C) − cV(C) for C ∈ C, where c = φ([0, 1] d ). ψ is a valuation which satisfies the
assumptions in (8). Thus ψ = 0, i.e. φ = cV, concluding the proof of the theorem.
⊓⊔
Remark. If, in this result, the group of proper rigid motions is replaced by the group
of translations, then, besides the valuations of the form cV, many other valuations
turn up. The situation on C thus is essentially more complicated than the situation on
B, see Theorem 7.6.
Hadwiger’s Functional Theorem
The above characterizations of the volume will lead to an easy proof of Hadwiger’s
[462, 464, 468] celebrated functional theorem. A preliminary version of this result is
due to Blaschke [128], Sect. 4, who also provided the basic idea of proof. Unfortunately,
Blaschke’s presentation was rather sloppy and it needed the genius of
Hadwiger to recognize this treasure.