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
= x
Rd
: (x) = τ .
The hyperplane 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 be the restriction of f onto . Thus
if f =
m
i=1
αi[Ai], then =
m
i=1
αi[Ai ]
and so K(Hτ ) and we can define χτ (fτ ). Since Ai 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 = =⇒ 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.
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