1. Convex Sets 3 (1.3) Some interesting examples. Sometimes, it is very easy to see whether the set is convex or not (see Figure 1). convex convex non-convex Figure 1. Two convex and one non-convex set Sometimes, however, this is not so easy to see (cf. Problems 3, 4 and 5 below), or a convex set may have a number of equivalent descriptions and their equivalence may be not obvious (cf. Problems 6 and 7 below). PROBLEMS. We will encounter many of the harder problems later in the text. 1◦. Prove that the convex hull of a set is a convex set. 2◦. Let c1,... , cm be vectors from Rd and let β1,... , βm be numbers. The set A = x ∈ Rd : ci,x≤ βi for i = 1, . . . , m is called a polyhedron. Prove that a polyhedron is a convex set. 3. Let v1,... , vm ∈ Rd be points. Let us fix positive numbers ρ1,... , ρm and let us define a map: H : Rd −→ Rd by H(x) = 1 f(x) m i=1 ρi exp x, vi vi, where f(x) = m i=1 ρi exp x, vi . a◦) Prove that the image of H lies in the convex hull of v1,... , vm. b∗) Prove that the image of H is convex and that for 0 and for any y ∈ conv ( v1,... , vm ) there exists an x ∈ Rd such that dist (any H(x),y ) .

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