Gruber P. Convex and Discrete Geometry

126 Convex Bodies
7.3 Characterization of Volume and Hadwiger’s Functional Theorem
Elementary volume on B and volume on C are valuations with particular properties,
including simplicity, translation and rigid motion invariance, monotonicity and
continuity. The valuation property and the specified properties are what one would
expect from a notion of volume in an axiomatic theory. Thus it is a natural question
to ask, whether the valuation property together with some of the specified properties
characterize elementary volume on B and volume on C.
In this section we give positive answers to the above questions. As an application,
we prove Hadwiger’s functional theorem, which by some mathematicians, for
example by Rota, is considered to be one of the most beautiful and interesting theorems
of all mathematics. Of its numerous applications, one in integral geometry is
presented, the principal kinematic formula, see Theorem 7.10.
In Sect. 16.1 we will show that a valuation on the space P of convex polytopes,
with certain additional properties, is a multiple of the elementary volume on P.
A result similar to the functional theorem, but for valuations on the space of lattice
polytopes, is the Betke–Kneser theorem 19.6.
We follow Klain, see [586]. A different modern proof of Hadwiger’s volume
theorem is due to Chen [203].
Characterization of the Elementary Volume of Boxes
As a first characterization theorem we show the following result.
Theorem 7.6. Let φ be a simple, translation invariant valuation on B which is
monotone or continuous. Then φ = c V , where c is a suitable constant.
Proof. Let c = φ([0, 1] d ). φ is a simple, translation invariant valuation on B and satisfies
the inclusion–exclusion formula, by Volland’s polytope extension theorem 7.2.
Since, for each n = 1, 2,..., the unit cube [0, 1] d can be dissected into n d cubes,
each a translate of the cube [0, 1 n ]d , it follows that
φ �� 0, 1 �d� 1
=
n nd φ� [0, 1] d� = c
nd = cV��0, 1 �d� for n = 1, 2,...
n
This, in turn, implies that
φ(B) = cV(B) for each box B with rational edge-lengths,
on noting that each such box can be dissected into translates of the cube [0, 1 n ]d for
suitable n. The monotonicity, respectively, the continuity then yields
φ(B) = cV(B) for each B ∈ B. ⊓⊔
Example. The assumption that φ is monotone or continuous on B is essential, as
the following valuation ψ on B(E 1 ) shows: Let f : R → R be a non-continuous
solution of Cauchy’s functional equation