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.

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