Gruber P. Convex and Discrete Geometry

130 Convex Bodies
Since the Riemann sums (actually functions of u) of the above parameter integrals
for the support functions hY = C f − , h Z = C f + are linear combinations of support
functions of line segments and thus of zonotopes, and since the Riemann sums converge
uniformly to hY and h Z , it follows that Y and Z are zonoids. ⊓⊔
The following tool is due to Sah [873].
Lemma 7.3. Let S be a d-dimensional simplex. Then S can be dissected into finitely
many convex polytopes, each symmetric with respect to a hyperplane.
Proof. Let F1,...,Fd+1 be the facets of S and let c ∈ S be the centre of the unique
inball of maximum radius of S. Letpi be the point where the inball touches Fi. For
i < j let Pij = conv{c, pi, p j, Fi ∩ Fj}. Then each Pij is a convex polytope, symmetric
in the hyperplane through c and Fi ∩ Fj, and {P12,...,Pdd+1} is a dissection
of S. ⊓⊔
Proof of the Theorem. The main step of the proof is to show the following proposition:
(8) Let ψ be a simple, continuous valuation on C which is invariant with respect
to proper rigid motions and such that ψ([0, 1] d ) = 0. Then ψ = 0.
We prove (8) by induction on d.Ifd = 1, then ψ is simple and translation invariant.
Thus ψ([0, 1]) = 0 implies that ψ([0, 1 n ]) = 0forn = 1, 2,... This, in turn,
shows that ψ vanishes on all compact line segments with rational endpoints. By continuity,
ψ then vanishes on all compact line segments, that is, on C(E1 ), concluding
the proof of (8) in case d = 1.
Assume now that d > 1 and that (8) holds in dimension d − 1. The proof of
Proposition (8) for d is divided into a series of steps.
First, Volland’s polytope extension theorem 7.2, applied to the valuation ψ|P,
shows that ψ satisfies the inclusion–exclusion formula on P and thus, being simple,
(9) ψ is simply additive on P.
Second,
(10) ψ(S) = ψ(−S) for each simplex S.
If d is even, S can be transformed into −S by a proper rigid motion and (9) holds
trivially. If d is odd, there is a dissection {P1,...,Pn} of S by Lemma 7.3 where each
Pi is a convex polytope which is symmetric with respect to a hyperplane. Hence Pi
can be transformed into −Pi by a proper rigid motion. By the assumption in (8), ψ is
invariant with respect to proper rigid motions. This together with (9) then yields (10).
Third,
(11) ψ(W ) = 0 for each right cylinder W ∈ C.
Embed E d−1 into E d = E d−1 × R as usual and define a function µ : C(E d−1 ) →
R by
µ(C) = ψ(C ×[0, 1]) for C ∈ C(E d−1 ) ⊆ C(E d ).