Gruber P. Convex and Discrete Geometry

7 Valuations 129
Theorem 7.8. Let φ be a simple, continuous valuation on C which is invariant with
respect to proper rigid motions. Then φ = cV , where c is a suitable constant.
The following proof is due to Klain [586], see also [587]. It simplifies the elementary
original proof of Hadwiger, but uses the non-elementary tool of spherical harmonics.
A recent simplified elementary version of the original proof of Hadwiger is due to
Chen [203].
First, some tools are put together. A convex polytope Z is a zonotope if it can be
represented in the form
Z = S1 +···+Sn, where the Si are line segments.
A convex body which is the limit of a convergent sequence of zonotopes is a zonoid.
For more on zonotopes and zonoids, see Goodey and Weil [384].
Lemma 7.2. Let C be an o-symmetric convex body with support function hC such
that the restriction of hC to S d−1 is of class C ∞ . Then there are zonoids Y and Z
such that
C + Y = Z.
Proof. If f : Sd−1 → R is a function of class C∞ , its cosine transform C f : Sd−1 → R
is defined by
�
C f (u) = |u · v| f (v) dσ(v)for u ∈ S d−1 ,
S d−1
where σ is the ordinary surface area measure in Ed . Using the Funk–Hecke theorem
on spherical harmonics it can be shown that C is a bijective linear operator on the
space of all even functions f : Sd−1 → R of class C∞ onto itself. See, e.g. [402].
Since hC|S d−1 is even and of class C∞ by assumption, there is an even function
f : Sd−1 → R of class C∞ such that hC|S d−1 = C f , that is,
�
hC(u) = |u · v| f (v) dσ(v)for u ∈ S d−1 .
S d−1
Define f + , f − : Sd−1 → R by f + (v) = max{ f (v), 0}, f − (v) = max{− f (v), 0}
for v ∈ Sd−1 . Then f = f + − f − and thus
�
hC(u) + |u · v| f − �
(v) dσ(v) = |u · v| f + (v) dσ(v)for u ∈ S d−1 .
S d−1
S d−1
It is easy to check that the cosine transforms C f − , C f + (homogeneously extended
to E d of degree 1) are positive homogeneous of degree 1 and convex. Thus the characterization
theorem 4.3 for support functions shows that there are convex bodies
Y, Z such that hY = C f − , h Z = C f + and therefore hC + hY = h Z ,or
C + Y = Z.