POLARS AND DUAL CONES

Transcription

1 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 R n is called convex if for any x, y C the point λx+(1 λ)y also belongs to C for any λ [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 R n is denoted by [x, y], so we have Likewise, we can consider open line segments [x, y] = (1 α)x + αy, α [0, 1]. (x, y) = (1 α)x + αy, α (0, 1). 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 by means of a convex combination. Given a finite set x 1, x 2,..., x p R n, a point x R n is the convex combination of x 1, x 2,..., x p if it can be represented as p p x = α i x i, α i 0 i = 1,..., p, α i = 1. Given an arbitrary set S R n, we can consider all possible finite convex combinations of points from this set. 1

2 2 VERA ROSHCHINA Definition 2 (Convex hull). The convex hull co S of a set S R n is the set of all convex combinations of finite subsets of S, p p co S = x = α i x i p N, xi S, α i 0 i = 1,..., p, α i = 1. Example 1. A standard simplex in R n is the convex hull of the coordinate vectors (see Fig. 3). x 2 (0,1) x 3 (0,0,1) (0,1,0) x 2 (1,0) x 1 (1,0,0) x 1 Figure 3. Standard simplices 2 and 3. We have explicitly n = co e 1,..., e n = α i e i αi 0 i = 1,..., n, α i = 1 = (x 1,..., x n ) x i 0 i = 1,..., n; where by e i, i 1,..., n we denote the coordinate vectors: x i = 1, e 1 = (1, 0,..., 0), e 2 = (0, 1, 0,..., 0),..., e n = (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 x i = 1, y i = 1, x i, y i 0 i 1,..., n. Therefore, for any λ [0, 1] and z = (1 λ)x + λy z i = [(1 λ)x i + λy i ] = (1 λ) x i + λ y i = (1 λ) + λ = 1, and at the same time z i = (1 λ)x i + λy i 0, so z 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 R n is a convex set. Proof. Let x, y co S. By the definition of a convex hull, we have the representations r s x = α i x i, y = β j y j, j=1

3 POLARS AND DUAL CONES 3 where α i s and β j s are the relevant convex coefficients, and x i, y j S for i = 1,..., r and j = 1,..., s. For z = µx + (1 µ)y with µ [0, 1] we have r s r s (1) z = µ α i x i + (1 µ) β j y j = µα i x i + (1 µ)β j y j, j=1 γ j=1 i γ j+r so we can define new coefficients γ 1,..., γ r+s as follows It is not difficult to observe that γ i := µα i i = 1,..., r; γ j+r := (1 µ)β j, j = 1,..., s. γ i 0 i 1,..., r + s and r+s γ i = 1, hence, (1) is a representation of z as the convex combination of r + s elements from S, and we therefore have z 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. polytopes (see Fig. 4). Platonic solids are classic examples of Figure 4. Platonic solids. The convex hull of S R n 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 R n, then co S is the intersection of all convex sets in R n 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 co S \ C. Since x co S, we have (2) x = α 1 x 1 + α 2 x α p x p, where x i S C for i 1,..., p and α 1,..., α p are convex combination coefficients: p α i = 1, α i 0 i 0,..., p. Without loss of generality we can assume that α i 0 for all i 1,..., p (otherwise we discard the points with zero coefficients). If p = 1, then x = 1 x S C. If p = 2, then x = (1 λ)x 1 + λx 2 C by the convexity of C. It remains to consider the case when p > 2. Let α := α p 1 (0, 1), α p 1 + α p and define a new quantity x p 1 := α x p 1 + (1 α )x p = α p 1 α p x p 1 + x p C α p 1 + α p α p 1 + α p

4 4 VERA ROSHCHINA by the definition of convexity. Also note that α p 1 x p 1 + α p x p = (α p 1 + α p )x p 1, and letting α p 1 = α p 1 + α p we can rewrite (2) as a convex combination of p 1 points from C: x = α 1 x 1 + α 2 x 2 + α p 1x p 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 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 R n. 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 R n. 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 R n be a nonempty closed convex set, and let z R n be such that z / C. Then there exists u S n 1 and x 0 C such that (3) u, z x 0 > 0; u, x x 0 0 x C, i.e. the point z can be strictly separated from the set C by a hyperplane. Here we use the standard inner product x, y = x T y. 2. Polars of convex sets Definition 5 (Polar). Let C R n be a convex set. Its polar is C = s R n s, x 1 x C. 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 = x x 1. The unit ball is self polar or self dual, i.e. B = B. To see this, we can use the definition directly, so (4) B = y R n y, x 1 x B = y R n sup y, x 1 x B. We have from the Cauchy-Schwarz inequality (5) x, y x y 1 y = y x B, on the other hand, for y 0 and x = y/ y we have (6) x, y = The equations (5) and (6) together yield y, y y = y sup y, x = y, x B and from (4) we have B = y R n sup y, x 1 x B = y y 1 = B.

5 POLARS AND DUAL CONES 5 Example 3. The polar of a cube centred at zero is an octahedron. Consider the cube We have explicitly hence, C = (x 1, x 2, x 3 ) 1 x i 1 i 1 : 3. sup y, x = sup (x 1 y 1 + x 2 y 2 + x 3 y 3 ) = y 1 + y 2 + y 3 = y 1, x C 1 x i 1 C = y R n sup y, x 1 x C = y R n y 1 1, an octahedron bounded by the 8 planes ±y 1 ± y 2 ± y 3 1. Observe that if we make the cube larger or smaller, so C α := (x 1, x 2, x 3 ) α x i α i 1 : 3 for some positive α, the same kind of computation yields C = y R n y 1 α 1. Note that the polars of other Platonic solids are also (appropriately positioned and scaled) Platonic solids. Next we will learn how to construct polars of arbitrary convex sets. Lemma 2. Let a closed convex set C be defined as the intersection of half-spaces, C = x R n x, a t b t > 0, t T, where T is an arbitrary index set, not necessarily finite. Then C = cl co (a t /b t, t T 0), where cl co is the closed convex hull (closure of the convex hull). Proof. Denote D := cl co (a t /b t, t T 0). Observe that for every x C by the definition of C we have a t /b t, x 1, hence, a t /b t C. Also obviously 0 C. Since C is convex and closed, D C. To show the reverse inclusion, assume the contrary: there is z 0 C \ D. By the separation theorem (Theorem 2) there exists u R n such that u, z 0 > sup u, x 0, x D the last inequality due to 0 D. Without loss of generality (scaling u if necessary), we have u, z 0 > 1 sup u, x 0. x D Observe that 1 sup u, x = max0, sup a t /b t, u, x D t T therefore a t, u b t t T, and hence u C. On the other hand, we have z 0 C and u, z 0 > 1, which is impossible. This contradiction finishes the proof. In view of Lemma 2 it is evident that dodecahederon and icosahedron are polars of each other. We will consider another example.

6 6 VERA ROSHCHINA Example 4. Let C R 3 be a cylinder, We will show using Lemma 2 that C = (x 1, x 2, x 3 x x 2 2 1, 1 x 3 1. C = co(0, 0, 1), (0, 0, 1), (x 1, x 2, 0), x x 2 2 = 1, i.e. C is the convex hull of a circle located at x 3 = 0 and two points (0, 0, 1) and (0, 0, 1), see Fig. 5. Figure 5. A cylinder and its polar First we need to obtain the explicit representation of the cylinder C in terms of a t and b t. The last constraint 1 x 3 1 translates into two inequalities x 3 1, x 3 1, so we can let a 0 1 = (0, 0, 1), a 0 2 = (0, 0, 1), b 0 1 = b 0 2 = 1. The inequality x x can be equivalently written as the system of linear inequalities From here we have the second family By Lemma 2 we have x 1 y 1 + x 2 y 2 1 y, y = 1. (a t, b t ) = (t, 1), t = (t 1, t 2 ), t t 2 2 = 1. C = cl co0 3, a 0 1, a 0 2, a t, t T = co(0, 0, 1), (0, 0, 1), (t 1, t 2, 0), t t 2 2 = 1. This is the convex hull of the circle x x 2 2 = 1 and the two points (0, 0, 1) and (0, 0, 1). We can consider the operation of taking polars of polars, so we can have a bipolar C. We summarise the properties of polars in the next statement (see [3, Theorem 2.11]). Theorem 3 (Elementary properties of polars). Let P, Q R n. We have the following properties of their polars: (i) P Q implies P Q and P Q, (ii) P P, (iii) P is convex and closed, (iv) 0 P.

7 Lemma 3 (Double Polar). For any C R n one has POLARS AND DUAL CONES 7 C := cl co0 C. Proof. Observe that C is a closed convex set. We will first show that (7) cl co0 C C. Indeed, and for all x C 0, y 1 y C, x, y 1 y C by the definition of polar, and we deduce (7) first by observing that the inequalities hold for the convex hull, and then for the closure of this convex hull. It remains to show that C cl co0 C. Assume that this is not true. Then there exists z C \ cl co0 C. We can separate z strictly from cl co0 C by the separation theorem. There is a vector u such that u, z > sup x cl co0 C u, x. Observe that the supremum on the right hand side is nonnegative (because it is taken over a set that contains zero), hence, w.l.o.g. we can assume (by scaling u) that u, z > 1 sup u, x. x cl co0 C By the definition of polar we have u C, but then z / C, which contradicts the assumption. 3. Dual cones Definition 6 (Cone). A set K R n is called a cone if for any point x K the ray is a subset of K. R ++ x = λx λ > 0 Cones are collections of rays emanating from the origin (that may not include the origin). A convex cone is a cone that is also a convex set. When K is a cone, its polar is a cone as well, and we can write (8) K = s R n s, x 0 x K, i.e. in the definition one can be replaced by zero. The equivalence is not difficult to see from the fact that K is a cone. Let us note some straightforward properties. The set K is a closed convex cone. If K is a subspace, then K coincides with its orthogonal complement, K = K. The only possible element in K K is 0. We can compute the double polar of a cone K by taking the polar of the polar to obtain K. This set is called a bipolar cone of K. It does not make sense to consider any further polars of polars due to the following statement. Lemma 4. If K is a nonempty convex cone, then K = cl K. Proof. Follows from Lemma 3 observing that 0 cl K.

8 8 VERA ROSHCHINA Polars of polyhedral cones have a particularly convenient and explicit representation. We have the following result. Lemma 5. Let K be a polyhedral cone, i.e. Then K = x x, a i 0 Proof. Follows directly from the representation and the separation argument. i 1,..., m. K = conea 1,..., a m. K = s R n s, x 0 x K Definition 7. Let K R n be a closed convex cone. Its dual cone is defined as K = u R n x, u 0 x K. It follows directly from the definition that the dual of a linear subspace is its orthogonal complement. Also note that K = K for any convex cone K. Definition 8. A closed convex cone K R n is called self-dual if K = K. Some notable examples of self-dual cones are the nonnegative orthant R n + = x x i 0 i 1,..., n, the second-order cone L n = x = ( x, x 0 ) R n R x x 0 and the cone of positive semidefinite matrices S n +. References [1] Jean-Baptiste Hiriart-Urruty and Claude Lemaréchal. Fundamentals of convex analysis. Grundlehren Text Editions. Springer-Verlag, Berlin, Abridged version of ıt Convex analysis and minimization algorithms. I and II. [2] R. Tyrrell Rockafellar. Convex analysis. Princeton Mathematical Series, No. 28. Princeton University Press, Princeton, N.J., [3] Günter M. Ziegler. Lectures on polytopes, volume 152 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995.