Gruber P. Convex and Discrete Geometry

Extension of Valuations on Simplices
7 Valuations 117
Ludwig and Reitzner [667] showed that each valuation on the space of all simplices
in E d has unique extensions to the spaces P and L(P) of all convex, respectively, all
convex and polyconvex polytopes. This result yields an alternative way to define the
elementary volume of polytopes, see Sect. 16.1.
Extension of Continuous Valuations on Convex Bodies
The following problem still seems to be open.
Problem 7.1. Can every valuation on C be extended to a valuation on L(C), the
space of polyconvex bodies?
While the general extension problem is open, in important special cases, fortunately,
the extension is possible. Considering the above proof of Volland’s theorem, it is
plausible that continuous valuations on C can be extended to valuations on L(C),
where continuity is meant with respect to the topology induced by the Hausdorff
metric. In fact, this is true, as the following extension theorem of Groemer [402]
shows.
Theorem 7.3. Let φ be a continuous valuation on C. Then φ can be extended
uniquely to a valuation on L(C).
Proof. By Theorem 7.1 it is sufficient to show that φ satisfies the inclusion–exclusion
formula (2). Let C, C1,...,Cm ∈ C such that C = C1 ∪···∪Cm.
First, the following will be shown:
(16) Let P1,...,Pm ∈ P such that C1 ⊆ int P1,...,Cm ⊆ int Pm. Then there
are Q1,...,Qm ∈ P such that C1 ⊆ Q1,...,Cm ⊆ Qm and Q1 ∪···∪
Qm ∈ P.
P = P1 ∪···∪Pm is a not necessarily convex polytope containing the convex body
C in its interior. Hence we may find a convex polytope Q with C ⊆ Q ⊆ P. Now
let Q1 = P1 ∩ Q,...,Qm = Pm ∩ Q.
By (16) we may choose m decreasing sequences of convex polytopes, say
(17) (Qin) ⊆ P such that Qi1 ⊇ Qi2 ⊇···→Ci as n →∞
for i = 1,...,m,
such that
Then
(18) Q1n ∪···∪ Qmn ∈ P.
(19) Q I 1 ⊇ Q I 2 ⊇···→CI for all I .