6 I. Convex Sets at Large In particular, if B = {b} is a point, the set A + b = x + b : x A is a translation of A. For a number α and a subset X V , the set αX = αx : x X is called a scaling of X (for α 0, the set αX is also called a dilation of X). Some properties of convex sets are obvious, some are not so obvious, and some are quite surprising. PROBLEMS. We will encounter some of the harder problems below later in the text. 1◦. Prove that the intersection i∈I Ai of convex sets is convex. 2◦. Let A V be a convex set and let T : V −→ W be a linear transformation. Prove that the image T (A) is a convex set in W . 3. Let A Rn be a polyhedron (see Problem 2, Section 1.3) and let T : Rn −→ Rm be a linear transformation. Prove that the image T (A) is a polyhedron in Rm. Remark: We prove this in Section 9 see Theorem 9.2. 4. Prove that A + B is a convex set provided A and B are convex. Prove that for a convex set A and non-negative numbers α and β one has (α+β)A = αA+βA. Show that the identity does not hold if A is not convex or if α or β are allowed to be negative. 5∗. For a set A Rd, let [A] : Rd −→ R be the indicator function of A: [A](x) = 1 if x A, 0 if x / A. Let A1,... , Ak be compact convex sets in Rn and let T : Rn −→ Rm be a linear transformation. Let Bi = T (Ai) be the image of Ai. Suppose that k i=1 αi[Ai] = 0 for some numbers αi. Prove that ∑k i=1 αi[Bi] = 0. Show that this is no longer true if the Ai are not convex. Remark: We prove this in Section 8 see Corollary 8.2. 6. Let A Rd be a compact convex set and B = (−1/d)A. Prove that there exists a vector b Rd such that b + B A. Remark: Figure 2 illustrates the statement for d = 2. We go back to this problem in Section 5 when we discuss Helly’s Theorem (see Problem 1 of Section 5.2 and the hint thereafter).
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