Gruber P. Convex and Discrete Geometry

7 Valuations 131
It is easy to see that µ is a simple, continuous valuation on C(Ed−1 ) which is invariant
with respect to proper rigid motions in Ed−1 and such that µ([0, 1] d−1 ) = 0. Thus
µ = 0onC(Ed−1 ) by the induction hypothesis. Hence ψ(W ) = 0 for all right
cylinders W of the form W = C ×[0, 1] ⊆Ed−1 ×R = Ed with basis C ∈ C(Ed−1 ).
Since ψ is invariant with respect to proper rigid motions in Ed , this yields (11) for
all right cylinders of height 1. Since ψ is simple and translation invariant, ψ(W ) = 0
for each right cylinder of height 1 n
, n = 1, 2,...,and thus for all right cylinders of
rational height. The continuity of ψ finally yields that (11) holds generally.
Fourth,
(12) ψ(X) = 0 for each slanting cylinder X ∈ C with polytope basis.
If X is long and thin, cut it with a hyperplane orthogonal to its cylindrical boundary
into two pieces and glue the pieces together such as to obtain a right cylinder YW.
Since ψ is translation invariant, (9) and (11) show that ψ(X) = ψ(W ) = 0. If X
is not long and thin, dissect it into finitely many long and thin slanting cylinders
X1,...,Xn, say. Since ψ satisfies (9), ψ(X) = ψ(X1) +···+ψ(Xn) = 0 by what
was just proved. This concludes the proof of (12).
Fifth,
(13) ψ(P + L) = ψ(P) for each P ∈ P and any line segment L.
Let L =[o, s] with s ∈ E d .IfdimP ≤ d − 1, then P + L is a cylinder or of
dimension ≤ d − 1. Using (12), respectively, the assumption that ψ is simple, we
see that ψ(P + L) = 0. Clearly, ψ(P) = 0 too, by the simplicity of ψ. Hence
ψ(P + L) = ψ(P) in case dim P ≤ d − 1. Assume now that dim P = d. Let
F1,...,Fn be the facets of P and denote the exterior unit normal vector of Fi
by ui. By renumbering, if necessary, we may suppose that s · ui > 0 precisely
for i = 1,...,m(< n). Then P, F1 + L,...,Fm + L form a dissection of P + L.
(9) and (12) then show that
ψ(P + L) = ψ(P) + ψ(F1 + L) +···+ψ(Fm + L) = ψ(P),
concluding the proof of (13).
Since a zonotope is a finite sum of line segments, a simple induction argument,
starting with (13), shows that ψ(P + Z) = ψ(P), ψ(Z) = 0 for each P ∈ P and
each zonotope Z. Since, by assumption, ψ is continuous, this implies that
(14) ψ(C + Z) = ψ(C), ψ(Z) = 0 for each C ∈ C and each zonoid Z.
Sixth,
(15) ψ(C) = 0 for each centrally symmetric convex body C ∈ C such that the
restriction of hC to S d−1 is of class C ∞ .
By Lemma 7.2 there are zonoids Y and Z such that C + Y = Z. Hence ψ(C) =
ψ(C + Y ) = ψ(Z) = 0 by (14), which yields (15). Since the centrally symmetric
convex bodies C with hC|S d−1 of class C ∞ are dense in the family of all centrally
symmetric convex bodies, (15) and the continuity of ψ imply that
(16) ψ(C) = 0 for all centrally symmetric C ∈ C.