24 I. Convex Sets at Large PROBLEMS. 1. Let S ⊂ Rd be a compact convex set. Prove that there is a point u ∈ Rd such that (−1/d)S + u ⊂ S. Hint: For every point x ∈ S consider the set Ax = u : (−1/d)x + u ∈ S . Use Helly’s Theorem. 2∗ (“Ham Sandwich Theorem”). Let μ1,... , μd be a set of Borel probability measures on Rd. Prove that there exists a hyperplane H ⊂ Rd such that μi(H+) ≥ 1/2 and μi(H−) ≥ 1/2 for all i = 1, . . . , d. 3∗ (Center Transversal Theorem). Let μ1,... , μk, k ≤ d, be Borel probability measures on Rd. Prove that there exists a (k − 1)-dimensional aﬃne subspace L ⊂ Rd such that for every closed halfspace H containing L we have μi ( H ) ≥ 1/(d − k + 2). Remark: For Problems 2 and 3, see [Z97] ˇ and references therein. 4◦. Prove that the theorem of Problem 3 implies both the result of Problem 2 above and the proposition of this section. A couple of geometric problems. 5∗ (Krasnoselsky’s Theorem). Let X ⊂ Rd be a set and let a, b ∈ X be points. We say that b is visible from a if [a, b] ⊂ X. Suppose that X ⊂ Rd is an infinite compact set such that for any d+1 points of X there is a point from which all d+1 are visible. Prove that there is a point from which all points of X are visible. 6 (Jung’s Theorem). For a compact set X ⊂ Rd, let us call maxy,z∈X y − z the diameter of X. Prove that any compact set of diameter 2 is contained in a ball of radius 2d/(d + 1). For Problems 5 and 6, see [DG63]. 6. An Application to Approximation We proceed to apply Helly’s Theorem to an important problem of constructing the best approximation of a given function by a function from the required class. We will go back to this problem again in Section IV.13. (6.1) Uniform approximations. Let us fix some real-valued functions fi : T −→ R, i = i, . . . , m, on some set T . Given a function g : T −→ R and a number ≥ 0, we want to construct a linear combination fx(τ) = m i=1 ξifi(τ), x = (ξ1, . . . , ξm) such that |g(τ) − fx(τ)| ≤ for all τ ∈ T. This is the problem of the uniform or Chebyshev approximation. Helly’s Theorem implies that a uniform approximation exists if it exists on every reasonably small subset of T .

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