26 I. Convex Sets at Large Proof. Let A(τ) : τ T be the sets defined in the proof of Proposition 6.2. Let A = A(σ1) . . . A(σn). First, we prove that A is compact. Indeed, by Problem 1, Section 6.2, the set A is closed. It remains to show that A is bounded. Let us define a function N : Rm −→ R, N(x) = max |fx(σi)| : i = 1, . . . , n . Then N(λx) = |λ|N(x) for λ R, N(x) 0 for x = 0 and N is continuous (in fact, N is a norm in Rm). Therefore, min N(x) : x = 1 = δ 0 and N(x) δ x . Now, if |g(σi) fx(σi)| for i = 1, . . . , n, we have |fx(σi)| |g(σi)| + for i = 1, . . . , n. Letting R = + max |g(σi)| : i = 1, . . . , n , we conclude that N(x) R, and, therefore, x≤ R/δ for any x A. Thus A is compact. For τ T let A(τ) = A(τ)∩A. Then each set A(τ) is compact. Applying Helly’s Theorem as in the proof of Proposition 6.2, we conclude that every intersection of a finite family of sets A(τ) is non-empty. Therefore, every intersection A(τ1) . . . A(τm+1) is a non-empty compact convex set. Therefore, By Corollary 4.3, the intersection of all the sets A(τ) is non-empty and so is the intersection of all the sets A(τ). A point x = (ξ1, . . . , ξm) τ∈T A(τ) gives rise to a function fx = ξ1f1 + . . . + ξmfm, which approximates g uniformly within the error . PROBLEMS. In the problems below, T = [0, 1] and fi(τ) = τ i , i = 0, . . . , m (note that we start with f0). 1◦. Prove that for any m + 1 distinct points τ1,τ2,... , τm+1 from [0, 1] the intersection A(τ1) . . . A(τm+1) is compact. 2◦. Let g(τ) = for τ [0, 1]. Let us choose = 0. Check that each intersection A(τ1)∩...∩A(τm+1) is not empty for any choice of τ1,... , τm+1 [0, 1], but τ∈[0,1] = ∅. In other words, for every m + 1 points τ1,... , τm+1 there is a
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