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Notes, We Review Useful Concepts in Convexity and Normed Spaces Mastermath, Spring 2019 Lecturers: D. Dadush, J. Briet Geometric Functional Analysis Lecture 0 Scribe: D. Dadush Review of Convexity and Normed Spaces In these notes, we review useful concepts in convexity and normed spaces. As a warning, many of the statements made here will not be proved. For detailed proofs, one is recommended to look up reference books in functional and convex analysis. The intent here is for these notes to serve as a reference for the basic language for the course. The reader may thus skim the material upon first reading, and then reread more carefully later during the course if the relevant concepts remain unclear. 1 Basic Convexity In this section, we will focus on convexity theorems in a finite dimensional real vector space, which will identify with Rn or subspaces theoreof. For vectors x, y 2 Rn, we define the standard T n n inner product hx, yi := x y = ∑i=1 xiyi. For two sets A, B ⊆ R and scalars c, d 2 R, we define the Minkowski sum cA + dB = fca + db : a 2 A, b 2 Bg. A convex set K ⊆ Rn is a set for which the line segment between any two points is in the set, or formally for which 8x, y 2 K, l 2 [0, 1], lx + (1 − l)y 2 K. A useful equivalent definition is that a non-empty set K is convex if 8a, b ≥ 0, aK + bK = (a + b)K. If K = −K, K is said to be symmetric. If K is compact with non-empty interior, then K is a convex body. We define the affine hull aff(K) to be the smallest affine subspace V containing K, and define the dimension dim(K) to be the dimension of its affine hull. We also define span(K) to be the smallest linear subspace containing K. Basic examples of convex sets, which will be important through the course, are the `p norm n n n p 1/p balls Bp = fx 2 R : kxkp ≤ 1g, where kxkp = (∑i=1 jxij ) , for p 2 [1, ¥) and kxk¥ = n n n maxi2[n] jxij. Note that B2 is the unit Euclidean ball and B¥ = [−1, 1] the n-dimensional cube. All the `p norm balls are symmetric convex bodies. 1.1 Representations of Convex Sets A first line of questioning for convex sets is that of representation. The two main representations we will interested in are in terms of convex hulls and linear inequalities, which are dual to each other. Given a set S ⊆ Rn, we define its convex hull by k k conv(S) = f∑ lixi : k 2 N, xi 2 S, i 2 [k], l ≥ 0, ∑ l = 1g . (1) i=1 i=1 It is easy exercise to check that for any convex set K, convex combinations of elements in K (e.g. k of the form ∑i=1 lixi as above) are also in K. It is also easy to see that conv(S) is convex by definition, thus conv(S) can be seen to be the smallest convex set containing S. In finite dimensions, it is useful to understand an effective upper bound on the maximum size of the convex combination k above needed to represent the convex hull. This is given by Caratheodory’s theorem: Theorem 1 (Caratheodory’s Theorem) For S ⊂ Rn and x 2 conv(S), there exists k ≤ n + 1, k k x1,..., xk 2 S, l1,..., lk ≥ 0, ∑i=1 li = 1, such that x = ∑i=1 lixi. 1 Thus in n dimensions, one requires a convex combination of at most n + 1 points, which is 1 1 easily seen to be tight: e.g. express ( n+1 ,..., n+1 ) as a convex combination of 0, e1,..., en, where e1,..., en are the standard basis vectors. Note that one can refine the above bound to d + 1, where d is the dimension of the affine hull of S. From the representation perspective, one may ask which convex sets can be expressed as convex hulls. In terms of the `p balls, it is a nice exercise to verify that n n n B1 = conv(±ei : i 2 [n]) B¥ = conv(f−1, 1g ) . For the above examples, we in fact have a minimimum size convex hull representation in terms of extreme points. A point v 2 K is an extreme point / vertex of a convex set K, if for 8x, y 2 K, l 2 (0, 1) (note the strict inclusion) lx + (1 − l)y = v ) x = y = v. That is, v cannot be expressed as a non-trivial convex combination of points in K. More generally, we say that F ⊆ K is a face of K, if F is convex and if 8x, y 2 K, l 2 (0, 1) lx + (1 − l)y 2 F ) x, y 2 F. Thus the extreme points of K, which we denote by ext(K), are precisely zero dimensional faces of K. The following theorem gives a useful answer to the convex hull representation question: Theorem 2 (Extreme Point Representation) For a compact convex set K ⊆ Rn, we have that K = conv(ext(K)). Note that for general closed convex sets, extreme points are not sufficient: e.g. [0, ¥) is not the convex hull of f0g. In this case, one also needs the concept of extreme rays, which we do not delve into now. Compact convex sets with a finite number of extreme points are called polytopes. Note that n n n B1 , B¥ are indeed polytopes, however the Euclidean ball B2 is not. Indeed, it is easy to check n n−1 n−1 n n ext(B2 ) = S , where S := fx 2 R : kxk2 = 1g is the unit Euclidean sphere in R . A second form of representation is in terms of linear inequalities. The key concept here is that of separation, that is that points outside a convex set can be separated from it via a hyperplane. More generally, so can non-intersecting convex sets. We state this theorem below: Theorem 3 (Separation Theorem) Let A, B ⊆ Rn be non-empty convex sets such that A \ B = Æ. Then there exists y 2 Rn such that 1. Weak Separation: supfhy, xi : x 2 Ag ≤ inffhy, xi : x 2 Bg 2. Non-triviality: inffhy, xi : x 2 Ag < supfhy, xi : x 2 Bg Furthermore, if A is closed and B is compact, then y can be chosen so that (1) holds with a strict inequality. Firstly, note that the non-triviality requirement guarantees that y 6= 0. Importantly, the the- orem doesn’t guarantee that we can put a hyperplane “strictly between” A and B in general, which is known as strong separation. This indeed may be impossible. As an example, take A = f(0, y) : y ≥ 0g (y-axis) and B = f(x, y) : y ≥ 1/x, x > 0g. Here the only non-trivial sep- arating hyperplane is H = f(x, y) : x = 0, y 2 Rg, where the points in A satisfy x ≤ 0 and the points in B satisfy x ≥ 0, where H is at distance 0 from both A and B. Such a strictly 2 separating hyperplane can be found when A, B satisfy the additional conditions (as in the fur- thermore above), where we can now choose H = fx 2 Rn : hy, xi = tg, for any t satisfying supfhy, xi : x 2 Ag < t < inffhy, xi : x 2 Bg. From the above, one can easily derive that closed convex sets can be expressed as the intersec- tion of halfspaces. To state this, we first define the so-called support function. For a non-empty n n set A ⊆ R , we define the support function hA : R ! R [ f¥g of A, to be hA(y) = supfhx, yi : x 2 Ag . (2) n An important relation is that if A ⊆ B, then hA(y) ≤ hB(y) for all y 2 R , that is larger sets have larger support function. We may now state and prove the second main representation theorem: Theorem 4 (Intersection of Halfspaces) Let K ⊆ Rn be a non-empty closed convex set. Then n K = \y2Rn fx 2 R : hy, xi ≤ hK(y)g Proof: Note that K is contained in the RHS by definition, so it suffices to show that if z 2/ K then z is not in the RHS. By the separation theorem with A = K and B = fzg, noting that A is n closed and fzg is compact, there exists y 2 R such that hK(y) = supfhy, xi : x 2 Ag < hy, zi, as needed. 2 Convex sets representable as a finite intersection of halfspaces, i.e. for which the y’s above can be restricted to a finite set S ⊂ Rn, are known as polyhedra. It is direct to verify the polyhedral representations: n n n n n B1 = fx 2 R : hx, yi ≤ 1, 8y 2 f−1, 1g g B¥ = fx 2 R : −1 ≤ xi ≤ 1, 8i 2 [n]g . We note that a consequence of the Minkowski-Weil theorem is that P is a bounded polyhedron iff P has a finite number of extreme points. 1.2 Functionals, Polars and Norms The support function defined above is an important example of a convex function. A function f : D ! R [ f¥g is convex, if its domain D is a convex set, and for all x, y 2 D, l 2 [0, 1], f (lx + (1 − l)y) ≤ l f (x) + (1 − l) f (y). It is a easy exercise to check hA is indeed convex over n n R , for any non-empty set A ⊆ R .
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