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POLARS and DUAL CONES 1. Convex Sets

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POLARS and DUAL CONES 1. Convex Sets POLARS AND DUAL CONES VERA ROSHCHINA Abstract. The goal of this note is to remind the basic definitions of convex sets and their polars. For more details see the classic references [1, 2] and [3] for polytopes. 1. Convex sets and convex hulls A convex set has no indentations, i.e. it contains all line segments connecting points of this set. Definition 1 (Convex set). A set C ⊆ Rn is called convex if for any x; y 2 C the point λx+(1−λ)y also belongs to C for any λ 2 [0; 1]. The sets shown in Fig. 1 are convex: the first set has a smooth boundary, the second set is a Figure 1. Convex sets convex polygon. Singletons and lines are also convex sets, and linear subspaces are convex. Some nonconvex sets are shown in Fig. 2. Figure 2. Nonconvex sets A line segment connecting two points x; y 2 Rn is denoted by [x; y], so we have [x; y] = f(1 − α)x + αy; α 2 [0; 1]g: Likewise, we can consider open line segments (x; y) = f(1 − α)x + αy; α 2 (0; 1)g: and half open segments such as [x; y) and (x; y]. Clearly any line segment is a convex set. The notion of a line segment connecting two points can be generalised to an arbitrary finite set n n by means of a convex combination. Given a finite set fx1; x2; : : : ; xpg ⊂ R , a point x 2 R is the convex combination of x1; x2; : : : ; xp if it can be represented as p p X X x = αixi; αi ≥ 0 8i = 1; : : : ; p; αi = 1: i=1 i=1 Given an arbitrary set S ⊆ Rn, we can consider all possible finite convex combinations of points from this set. 1 2 VERA ROSHCHINA Definition 2 (Convex hull). The convex hull co S of a set S ⊆ Rn is the set of all convex combinations of finite subsets of S, ( p p ) X X co S = x = αixi p 2 N; xi 2 S; αi ≥ 0 8i = 1; : : : ; p; αi = 1 : i=1 i=1 Example 1. A standard simplex in Rn is the convex hull of the coordinate vectors (see Fig. 3). x2 x3 (0,0,1) (0,1) (0,1,0) x2 (1,0,0) (1,0) x1 x1 Figure 3. Standard simplices ∆2 and ∆3. We have explicitly ∆n = co fe1; : : : ; eng ( n n ) X X = αiei αi ≥ 0 8 i = 1; : : : ; n; αi = 1 i=1 i=1 ( n ) X = (x1; : : : ; xn) j xi ≥ 0 8 i = 1; : : : ; n; xi = 1 ; i=1 where by ei, i 2 f1; : : : ; ng we denote the coordinate vectors: e1 = (1; 0;:::; 0); e2 = (0; 1; 0;:::; 0); : : : ; en = (0;:::; 0; 1): It is not difficult to observe that the standard simplex is a convex set. Indeed, for every two points x and y in ∆n we have n n X X xi = 1; yi = 1; xi; yi ≥ 0 8 i 2 f1; : : : ; ng: i=1 i=1 Therefore, for any λ 2 [0; 1] and z = (1 − λ)x + λy n n n n X X X X zi = [(1 − λ)xi + λyi] = (1 − λ) xi + λ yi = (1 − λ) + λ = 1; i=1 i=1 i=1 i=1 and at the same time zi = (1 − λ)xi + λyi ≥ 0, so z 2 ∆n. This proof can be generalised to show that the convex hull of any set is convex. Lemma 1. The convex hull co S of any set S ⊆ Rn is a convex set. Proof. Let x; y 2 co S. By the definition of a convex hull, we have the representations r s X X x = αixi; y = βjyj; i=1 j=1 POLARS AND DUAL CONES 3 where αi's and βj's are the relevant convex coefficients, and xi; yj 2 S for i = 1; : : : ; r and j = 1; : : : ; s. For z = µx + (1 − µ)y with µ 2 [0; 1] we have r s r s X X X X (1) z = µ αixi + (1 − µ) βjyj = µαi xi + (1 − µ)βj yj; i=1 j=1 i=1 |{z} j=1 | {z } γi γj+r so we can define new coefficients γ1; : : : ; γr+s as follows γi := µαi i = 1; : : : ; r; γj+r := (1 − µ)βj; j = 1; : : : ; s: It is not difficult to observe that r+s X γi ≥ 0 8 i 2 f1; : : : ; r + sg and γi = 1; i=1 hence, (1) is a representation of z as the convex combination of r + s elements from S, and we therefore have z 2 co S by the definition of a convex hull. The result now follows from the arbitrariness of our choice of x, y and µ. The convex hull of a finite set is called a polytope. Platonic solids are classic examples of polytopes (see Fig. 4). Figure 4. Platonic solids. The convex hull of S ⊆ Rn can be defined equivalently as the smallest convex set containing S. To guarantee that such definition is correct, we need the following result. Theorem 1. Let S ⊆ Rn, then co S is the intersection of all convex sets in Rn containing S. Proof. First observe that S ⊆ co S, since for every point x in S we have x = 1 · x, a convex combination of one element. Furthermore, the set co S is convex by Lemma 1. We will show that any convex set that contains S also contains co S, and hence co S is the smallest convex set containing S. Assume the contrary. Then there exists a convex set C such that S ⊆ C and at the same time there is x 2 co S n C. Since x 2 co S, we have (2) x = α1x1 + α2x2 + ··· + αpxp; where xi 2 S ⊆ C for i 2 f1; : : : ; pg and α1; : : : ; αp are convex combination coefficients: p X αi = 1; αi ≥ 0 8 i 2 f0; : : : ; pg: i=1 Without loss of generality we can assume that αi 6= 0 for all i 2 f1; : : : ; pg (otherwise we discard the points with zero coefficients). If p = 1, then x = 1 · x 2 S ⊆ C. If p = 2, then x = (1 − λ)x1 + λx2 2 C by the convexity of C. It remains to consider the case when p > 2. Let α α0 := p−1 2 (0; 1); αp−1 + αp and define a new quantity 0 0 0 αp−1 αp xp−1 := α xp−1 + (1 − α )xp = xp−1 + xp 2 C αp−1 + αp αp−1 + αp 4 VERA ROSHCHINA by the definition of convexity. Also note that 0 αp−1xp−1 + αpxp = (αp−1 + αp)xp−1; 0 and letting αp−1 = αp−1 + αp we can rewrite (2) as a convex combination of p − 1 points from C: 0 0 x = α1x1 + α2x2 + ··· αp−1xp−1: We can continue this reduction process until the number of points in the representation is reduced to at most two, and conclude as before that x 2 C, which contradicts the assumption. Therefore, there is no smaller convex set than co S that contains C. Corollary 1. A set S is convex if and only if S = co S. Theorem 1 allows us to give two additional alternative definitions of a convex set. Definition 3 (Convex hull as the smallest convex set). Let S ⊆ Rn. The convex hull co S of the set S is the smallest convex set that contains S. Definition 4 (Convex hull as the intersection). Let S ⊆ Rn. The convex hull co S of the set S is the intersection of all convex sets that contain S. Separation theorem is one of the most important tools in convex geometry. We give the statement of this result without proof for now. Theorem 2 (Separation). Let C ⊆ Rn be a nonempty closed convex set, and let z 2 Rn be such that z2 = C. Then there exists u 2 Sn−1 and x0 2 C such that (3) hu; z − x0i > 0; hu; x − x0i ≤ 0 8x 2 C; i.e. the point z can be strictly separated from the set C by a hyperplane. Here we use the standard inner product hx; yi = xT y. 2. Polars of convex sets Definition 5 (Polar). Let C ⊆ Rn be a convex set. Its polar is ◦ n C = fs 2 R j hs; xi ≤ 1 8x 2 Cg: It is not difficult to observe that a polar of any set (not necessarily convex) is a closed convex set. Example 2. Consider the unit ball in the Euclidean space, B = fx j kxk ≤ 1g. The unit ball is self polar or self dual, i.e. B◦ = B. To see this, we can use the definition directly, so ◦ n n (4) B = fy 2 R j hy; xi ≤ 1 8x 2 Bg = y 2 R j suphy; xi ≤ 1 : x2B We have from the Cauchy-Schwarz inequality (5) hx; yi ≤ kxk · kyk ≤ 1 · kyk = kyk 8x 2 B; on the other hand, for y 6= 0 andx ¯ = y=kyk we have hy; yi (6) hx;¯ yi = = kyk kyk The equations (5) and (6) together yield suphy; xi = kyk; x2B and from (4) we have ◦ n B = y 2 R j suphy; xi ≤ 1 = fy j kyk ≤ 1g = B: x2B POLARS AND DUAL CONES 5 Example 3. The polar of a cube centred at zero is an octahedron. Consider the cube C = f(x1; x2; x3) j − 1 ≤ xi ≤ 1 8 i 2 1 : 3g: We have explicitly suphy; xi = sup (x1y1 + x2y2 + x3y3) = jy1j + jy2j + jy3j = kyk1; x2C −1≤xi≤1 hence, ◦ n C = y 2 R j suphy; xi ≤ 1 x2C n = fy 2 R j kyk1 ≤ 1g ; an octahedron bounded by the 8 planes ±y1 ± y2 ± y3 ≤ 1.
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