30 I. Convex Sets at Large
Let us prove that χ exists. First, we define χ on functions f ∈
K(Rd).
We use induction on d. Suppose that d = 0. Then any function f ∈
K(Rd)
has
the form f = α[0] for some α ∈ R and we let χ(f) = α.
Suppose that d 0.
For a point x = (ξ1, . . . , ξd), let (x) = ξd be the last coordinate of x. For a
τ ∈ R let us consider the hyperplane
Hτ = x ∈
Rd
: (x) = τ .
The hyperplane Hτ can be identified with Rd−1 and hence, by the induction hy
pothesis, there exists a valuation, say χτ : K(Hτ ) −→ R, which satisfies the required
properties. For a function f ∈
K(Rd),
let fτ be the restriction of f onto Hτ . Thus
if f =
m
i=1
αi[Ai], then fτ =
m
i=1
αi[Ai ∩ Hτ ]
and so fτ ∈ K(Hτ ) and we can define χτ (fτ ). Since Ai ∩ Hτ are compact convex
(possibly empty) sets, we must have
χτ (fτ ) =
i:Ai∩Hτ =∅
αi.
Let us consider the limit
lim
−→+0
χτ− (fτ− ).
It may happen that the limit is equal to χτ (fτ ). This happens, for example, if for
every i and small 0, we have Ai ∩ Hτ = ∅ =⇒ Ai ∩ Hτ− = ∅ (see Figure 8).
A
a
a
b
b


E
E
Figure 8. Example: for the function f = [A], we have
lim
−→+0
χa− (fa− ) = χa(fa) = 1 but 0 = lim
−→+0
χb− (fb− ) =
χb(fb) = 1.