Gruber P. Convex and Discrete Geometry

Then ψ, defined by
f (s + t) = f (s) + f (t) for s, t ∈ R.
ψ(B) = f (β − α) for B =[α, β] ∈B(E 1 ),
7 Valuations 127
is a simple, translation invariant valuation on B(E 1 ), but clearly not a multiple of the
length.
Remark. A similar result holds for L(B), where, by the continuity of a valuation on
L(B), we mean continuity of its restriction to B.
First Characterization of the Volume of Convex Bodies
In the following we present a characterization of the volume on C, see Hadwiger
[462,464,468]. Versions of this result in the context of Jordan and Lebesgue measure,
respectively, Riemann and Lebesgue integrals are well known.
Theorem 7.7. Let φ be a simple, translation invariant, monotone valuation on C.
Then φ = c V , where c is a suitable constant.
Proof. By replacing φ by −φ, if necessary, we may assume that
(1) φ is non-decreasing and thus non-negative on C.
An application of the above characterization theorem to the restriction of φ to B
shows that
(2) φ(B) = cV(B) for B ∈ B,
where c is a suitable constant. By Theorem 7.2 φ has a unique extension to a valuation
on L(B) which we also denote by φ. Similarly, cV is a valuation on L(B)
which extends the valuation cV on B, see Propositions 7.1, 7.2. Since φ and cV
coincide on B by (2), they must coincide on L(B) by Volland’s polytope extension
theorem 7.2:
(3) φ(A) = cV(A) for A ∈ L(B).
Volland’s theorem, applied to the restriction of the valuation φ to the intersectional
family P, shows that φ satisfies the inclusion–exclusion formula on P. Since
φ is simple, we see that
(4) φ is simply additive on P.
We now show that
(5) φ(A) ≤ φ(C) for A ∈ L(B), C ∈ C, where A ⊆ C.
(Note that this is not trivial since in the theorem monotonicity is assumed only for
C.) Given A, C, we have conv A ∈ P. Then A ⊆ conv A ⊆ C. Represent A in the
form
A = B1 ∪···∪ Bm, where Bi ∈ B