Real Analysis with Economic Applications - Chapter G pptx

Chapter G Con vexit y One major reason why linear spaces are so important for geom etric ana lysis is that they allow us to define the notion of “line segment” in algebraic terms. Am ong othe r things, this enables o ne to formulate, purely algebraically, the notion of “con vex set” whic h figures majorly in a variet y of bran ches of higher mathematics. 1 Immediately relevan t for economic theory is t he indispensable role play ed by convex sets in o p- timization theory. At least for econo mists, this alone is enough of a motiva tion for taking on a comprehensive study of convex sets and related concepts. We begin the c h ap te r with a fairly detaile d dis cu s sion of convex sets an d con es. Our emphasis is, again, on the infinite di mensional side of the picture. In particu- lar, we consider sev eral examples that are couched within infinite dimensional linear spaces. After all, one of our main objectives here is to provide some help for the novice to get over the sensation of shock that the strang e beha v ior of infinite dimensional spaces may inv oke at first. We also introduce the partially orde re d linea r spac es, and discuss the im portan t role played by conv ex cones thereof. Am ong the topics that are lik ely to be new to the reader are the algebr aic in te- rior and algebraic closure of subsets of a linear space. These notions are developed relatively leisu re ly, for t h ey are essential for th e tre atment of the high point s o f the c hap ter, nam ely, the fundamental extension and separation theorems in an arbitrary linear space. In particular, we pro ve here t he linear algebraic formulations of the Hahn-Banach Theorems on the extension of a linear functional and the separation of con vex sets, along with the Krein-Rutman Theor em on the e xtension of positiv e lin- ear functionals . We then turn to Eu clide an convex an alys is, and deduc e Min kowski’s Separating and Supporting Hyperplane Theorems — these are among the most widely used theorems in economic theory — as easy corollaries of our general results. As a final order of business, w e discuss the problem of best appro ximation from a con vex set in a Euclidea n space. In particular, we in troduce the orthogona l projection op- erator, and use it to obtain the Euclidean version of the K rein-Rutman T heorem. Several econ omic applications of the main theore ms established in this c h ap te r are cons idered in Chapter H. 2 1 See Klee (1971) for a concise, yet instructiv e, introduction to the theory of con vex sets at large. 2 There are many excellent textbooks on finite dimensional convex analysis. Rockafellar (1970) is a classic in the field, but of course, there a re more recent expositions. In particular, Hiriart-Urruty and Lemaréchal (2000) provide a very nice introduction, and Borwein and Lewis (2000) take one to the next leve l. Elementary treatments of infinite dimensional convex analysis are harder to find. Certainly the linear algebraic treatment I present here is not commonly adopted in textbooks on mathematical analysis. One major exception is the first chapter of the excellent text by H olm es (1975). However, if this is your first serious encounter with infinite dimensional linear s paces, I would suspect that it w ould be wiser to go to this reference only after com p leting the present chapter. 317 1ConvexSets 1.1 Basic D e finitions and Examples You are familiar with the w ay one defines a con vex set in the context of Euclidean spaces — we have already used this concept a few times in this book. Th is de finition carries ov er to the case of arbitrary linear spaces w ithout m odification. Dhilqlwlrq. For any 0 < λ < 1, asubsetS of a linear spac e X is said to be λ-convex if λx +(1−λ)y ∈ S for any x, y ∈ S, or equivalen tly, if λS +(1− λ)S = S. If S is 1 2 -convex, w e say that it is midpoint convex. Finally, if S is λ-con ve x for all 0 < λ < 1, that is, if λS +(1− λ)S = S for all 0 ≤ λ ≤ 1, then it is said to be a convex set. The i nterpretation of these conc epts are straigh tfo rward . For any two distinct vectors x and y in the linear s p a ce X, we think of the set {λx+(1−λ)y ∈ X : λ ∈ R} as thelinethroughx and y. From this point of view, the set {λx +(1−λ)y ∈ X : 0 ≤ λ ≤ 1} corresponds to thelinesegmentbetweenx and y. 3 Consequen tly, a subset S of X is midpoint conv ex iff it contains the m id point of the line segment bet ween an y t wo of its constituent vectors. It is convex iff it contains the entire line segment between any two of its elements (Figure 1). ∗∗∗∗FIGURE G.1 ABOUT HERE ∗∗∗∗ As for exam p le s, note that Q is a midpoint convex subset of R which is not convex. Indeed, a nonempty subset of R is convex iff it is an interval. In any linear space X, all singleton sets, line seg me nts and lines, alon g with ∅, are con v ex sets. An y linear subspace, affine ma n i fold , hyperplane or h alf space in X is also convex. For instance, for any 0 < λ < 1, ahyperplaneH in a linear space X is a λ-convex set, because, by Corollary F.4 , H = {x ∈ X : L(x)=α} fo r som e n onzer o L ∈ L(X, R) and α ∈ R, and th us, for an y x, y ∈ H, we have L(λx +(1−λ)y)=λL(x)+(1− λ)L(y)=λα +(1− λ)α = α, that is, λx +(1− λ)y ∈ H. 3 Observ e that it is the linear structure of a linear space that lets us “talk about” these objects — we could not do so, for instance, in an arbitrary metric space. 318 As an immed iate illu s tration of the appeal of convex s ets — w e will of cour se see man y more examples later — let u s note that the affin e h u ll of a con vex set can in general be expressed much more easily than that of a nonconvex set. Indeed, for any convex subset S of a linear space , we simply have aff (S)={λx +(1−λ)y : λ ∈ R and x, y ∈ S}. (Geometr ic interpretation?) The proof requires an easy induction argu men t — we leave that to you. But we note that convex ity is essential for this obse rvation. For instance, the set S := {0, e 1 , e 2 } in R 2 does not satisfy the equation above. Exercise 1. Determine which of the following sets are necessarily convex (in the linear spaces they naturally live): (a) a nonempty connected subset of R 2 ; (b) {(t, y) ∈ R 2 : f(t) ≥ y} where f is a concave self-map on R; (c) {(t, y) ∈ R 2 :sint ≥ y}; (d) an open neighborhood of a given point in R 4 ; (e) the set of all semicontinuous functions on [0, 1]; (f ) {f ∈ B[0, 1] : f ≥ 0}; (g) {f ∈ C[0, 1] : f(1) =0}. Exercise 2 . Sho w that the intersection of any collection of λ-convex subsets of a linear space is λ-convex, 0 < λ < 1. Exercise 3 . Show that if A and B are λ-convex subsets of a linear space, so is αA + B for any α ∈ R and 0 < λ < 1 . Exercise 4. H Prove: A convex set S is contained in one of the open halfspaces induced by a hyperplane H in a linear space iff S ∩H = ∅. Let S be an y set in a linea r space X, and let S be the class of all conv ex subsets of X that co ntain S. We have S = ∅ — after all, X ∈ S. Then , b y Exe rcis e 2, W S is aconvexsetinX whic h , obviously, contains S. Clear ly, this set is the smalle st (that is, ⊇-minimum) subset of X that contain s S — it is called the convex hull of S, and denote d by co(S). (Needless to say, S = co(S) iff S is con vex; see Figure 1.) This notion prov es useful when one needs the convexity of the s et o ne is working with, even though tha t set is n ot kno wn to be convex . We will encounter many situations of t his sort later. It is worth noting that the definition of co(S) uses v ectors that lie outside S (because the mem bers of S may w ell contain suc h v ecto rs). For this reason, this def- inition can be viewed as an external one. We may also characterize co(S) internally, that is, by using only the vectors in S. Indeed, co(S) is none other than the set of all con vex com binations of finitely man y mem bers of S. That is, 319 co(S)=  m S i=1 λ i x i : m ∈ N,x 1 , ,x m ∈ S and (λ 1 , , λ m ) ∈ m−1  (1) where  m−1 :=  (λ 1 , ,λ m ) ∈ [0, 1] m : m S i=1 λ i =1  ,m=1, 2, This shows that the way we define the conv ex hull of a set here is in full accord with how we defined this concept in Section E.5.3 to state Carathéodory’s Theorem in the context of Euclidean spaces. Moreo ver, for a ny x, y ∈ X, we see that co({x, y}) — which is denoted simply as co{x, y} henceforth — is nothing but the line segmen t between x and y, an d we ha ve co(S)= V {co(T ):T ∈ P(S)}, where P(S) is the class of all nonempty finite subsets of S. Exercise 5 . Prove (1). Exercise 6. Show that, for any subset S of a linear space X and x ∈ X, co(S + x)= co(S)+x. Exercise 7. Pro ve or disprove: If X and Y are linear spaces, and A ⊆ X and B ⊆ Y, then co(A × B)=co(A)× co(B), where the underlying linear space is X × Y (Exercise F.23). The definitions of concave and convex f u nctio ns ( Section A.4.5) also ex tend in the ob v ious way to the case of real ma p s defined on a convex subset of a linear space. Dhilqlwlrq. Let X be a linear spa c e and T a nonempty conv ex subset of X. Areal map ϕ on T is called concave if ϕ(λx +(1−λ)y) ≥ λϕ(x)+(1−λ)ϕ(y) for all x, y ∈ T and 0 ≤ λ ≤ 1, while it is called convex if −ϕ is concave. (If both ϕ and −ϕ are concav e, then ϕ is an affine map; recall Exercise F.32.) All algebraic properties of c on c ave f un ction s that you are familiar with are valid in this more general setup as well. For example, if T is a nonempty con vex subset of a linear space, then ϕ ∈ R T is concave iff {(x, a) ∈ T × R : ϕ(x) ≥ a} is a convex subset of T × R. Moreover, if ψ ∈ R T is also concave, then so is ϕ + ψ. How ev er, the set of all concave real funct ions on T is not an affine manifold of R T . (Why?) 320 Th e n ex t exercise introduces t he notion of convex correspondence, w hich plays an important role in optimization theory. In fact, we have already used con vex corre- sponden ces when studying the theory of dynamic program ming (recall Proposition E.7). We will come bac k to them in due course. Exercise 8. H (Convex Correspondences)LetX and Y be two linear spaces, and S a nonempty convex subset of X. We say that Γ : S ⇒ Y is a convex correspondence if Gr(Γ) is a convex subset of X × Y, that is, λΓ(x)+(1− λ)Γ(x  ) ⊆ Γ(λx +(1− λ)x  ) for any x, x  ∈ S and 0 ≤ λ ≤ 1. (a)LetT be a convex subset of Y ,takeanyL ∈ L(X, Y ), and define Γ : S ⇒ Y by Γ(x):=L(x)+T. Show that Γ is a convex correspondence. (b)Let f ∈ R S be a convex function. Show that Γ : S ⇒ R, defined by Γ(x):= {a ∈ R : a ≥ f(x)}, is a convex correspondence. (c) Show that the budget correspondence (Example E.2) is a convex-valued corre- spondence which is not convex. (d) Ev ery linear correspondence Γ : X ⇒ Y is convex (Exercise F.31). Is the con- verse true? (e) (Deutsch-Singer) Prove: If Γ : X ⇒ Y is a conv ex correspondence with |Γ(x 0 )| = 1 for some x 0 ∈ X, then there exists an affine map f ∈ Y X such that {f(x)} = Γ(x) for all x ∈ X. 1.2 Convex Cones Convex sets that are closed under nonnegative scalar multiplic ation pla y an important role in c onvex analysis and optimiza t ion theory. Let us giv e them a na me. Dhilqlwlrq. A nonempt y subset C o f a linear space X is said to be a cone if it is closed under nonnegative scalar multiplication, that is, λC ⊆ C for all λ ≥ 0, i.e. λx ∈ C fo r any x ∈ C and λ ≥ 0. If C is, in addition, closed under addition, that is, C + C ⊆ C, i.e. x + y ∈ C for any x, y ∈ C, then it is called a con vex cone.WesaythataconeC in X is pointed if C ∩−C = {0}, 4 generating if span(C)=X, and non trivial if C = {0}. Geometrically speaking, a cone is a set that contains all ra ys that start from the origin and pass t hr ough another member o f t h e set (Figure 2). I n turn, a convex 4 Reminder. −C := (−1)C, that is, −C = {−x : x ∈ C}, and C − C := C + −C, that is, C − C = {x − y : x, y ∈ C}. 321 cone is no ne other t han a cone which is also conv ex set. (Why?) In a manner o f speaking, the concept of convex cone lies in bet ween that of a convex set and that o f a linear subspac e. This point will become clear as we develop the theory of convex sets fur t her. Remark 1. Reca ll that th e dimen s ion of a s e t in a lin e a r space X is the dim e n sion of its affine hull (Remark F.1). B u t since an y cone C in X contains the origin 0 (yes?), we have aff (C)=span(C). (W hy?) Thus dim(C)=dim(span(C)), soaconeC in a finite dimensional linear space X is generating iff dim(C)=dim(X).  ∗∗∗∗FIGURE G.2 ABOUT HERE ∗∗∗∗ E{dpsoh 1. [1] Any linear subspace C of a linear space X is a con vex cone. In this case C ∩−C = C, so C is a pointed cone iff C = {0}. Moreover, C is generating iff C = X. [2] The sm a llest (i.e. ⊇-minimum) convex cone in a linear space X is the triv ial linear subspace {0} while the lar ge st (i.e. ⊇-maximum ) convex cone is X itself. Moreover, if C is a conv e x cone in th is space, so is −C. [3] For any given n ∈ N, the set R n + is a poin ted generating convex cone in R n — we have dim(R n + )=n. O n the other hand, R n ++ is not a cone in R n since it does not contain the origin 0. Indeed, R n ++ ∪{0} is a pointed generating convex cone in R n . An example of a convex cone in R n , n ≥ 2, which is not pointed i s {x ∈ R n : x 1 ≥ 0}, and that of one which is not generating is {x ∈ R n : x 1 =0}. (W hat a re the dimensions of t hese cones?) [4] If X and Y are t wo linear spaces, and L ∈ L(X, Y ), then {x ∈ X : L(x) ≥ 0} is a (poin ted) con vex cone in X. This observation allows us to find many convex cones. For instance, {(x m ) ∈ R ∞ : x 4 ≥ 0}, or {f ∈ B[0, 1] : f( 1 2 ) ≥ 0}, or {f ∈ C[0, 1] : U 1 0 f(t)dt ≥ 0} are pointed convex cones by t his very token. All o f these cones are infinite dimensional and generating. [5] Both {f ∈ C 1 [0, 1] : f ≥ 0} an d C := {f ∈ C 1 [0, 1] : f  ≥ 0} are conv ex cones in C 1 [0, 1]. The former one is pointed , b u t the latter is not — all constant functions belong to C ∩−C.  Let S be any nonem p ty set in a line ar spa ce X. It i s easily check e d that S is a convex cone in X iff S x∈T λ(x)x ∈ S for any nonempty finite subset T of S an d λ ∈ R T + . (Verify!) It follow s that the smallest convex cone that c o nta ins S —the conical hull of S — exists and equals the set of all positiv e linear combinations of finitely many mem bers of S. (Why?) D enotin g this con vex cone by cone(S), therefore, we ma y write cone(S)=  S x∈T λ(x)x : T ∈ P(S) an d λ ∈ R T +  322 where, as usual, P(S) stan ds for the class of all nonem p ty finite subsets of S.(By con ven tion, w e let cone(∅)={0}.) For instance, for any n ∈ N, cone(R n ++ )=R n ++ ∪{0} and cone({e 1 , , e n })=R n + , where {e 1 , , e n } is the standard basis for R n . Exercise 9. H Let c 0 be the linear space of all real sequences all but finitely many terms of which are zero. Prove: (a) C := {(x m ) ∈ c 0 : x m > 0 for some m} is not a cone in c 0 ; (b) C ∪ {0} is an infinite dimensional cone which is not midpoint convex; (c) cone(C)=c 0 . Exercise 10. Show that if C is a cone in a linear space X,thenspan(C)=C −C. Thus, C is generating iff X = C − C. Exercise 11. Let S be any nonempt y set in a given linear space. Show that cone(S)= V {λco(S):λ ≥ 0}. Exercise 12. H Let C be a cone in a given linear space. Prove or disprove: cone(C)= co(C). Let X be a linear space and C a nontrivial convex cone in X. A nonempt y convex subset B of C is called a base for C if, for each x ∈ C, there exists a unique (y,λ) ∈ B × R ++ such that x = λy. 5 (See Figure 3.) A lot can be learned about a convex cone with a base by studying the properties of its base alone. Since often a base for a convex cone is simpler to analyze than the cone itself, it is worth kno wing if a given cone has a base. The follow ing exercise pro v ides two criteria for this to be the case. ∗∗∗∗FIGURE G.3 ABOUT HERE ∗∗∗∗ ∗ Exercise 13. H Let C be a n ontrivial convex cone in a linear space X,andB a convex subset of X. Pro ve that the following statements are equivalent. (i) B is a base for C; (ii) C = V {λB : λ ≥ 0} and 0 /∈ aff (B); (iii) there exists an L ∈ L(X, R) such that L(C\{0}) ⊆ R ++ and B = {x ∈ C : L(x)=1}. 5 The uniqueness requirement is essential here. For instance, due to this requirement, no bas e can contain the origin 0 of X. For, if B is a nonempty convex subset of C with 0 ∈ B = {0}, then for any nonzero y ∈ B we also have 1 2 y ∈ B (because B is convex ), so y itself has t wo distinct representations: y =1y and y =2( 1 2 y). Thus in this case B cannot be a base for C. (The case B = {0} is trivially dismissed as a poten t ial candidate for a base, of course.) 323 1.3 Ordered Linear Spaces Let (X, ) be a preordered set (Section A.1.4). If X is a linear spa c e, and  is comp atible with the addition and scalar multiplicatio n operations on X in the sense that x  y if and on ly if λx + z  λy + z (2) for an y x, y, z ∈ X and λ ≥ 0, th en we s ay that (X,) is a preordered linear space, and refer to  as a v ector preorder. M ore precisely, a preordered linear space is a list (X, +, ·, ) where (X, +, ·) is a linear space , (X, )isapreorderedset, and th e se two mathema tical systems are connected via the co mp atib ility requirement (2). Of course, (X, +, ·, ) is unnecessarily mouthful. We denote this system simply as X to simp lify the notation, and write  X for the associated preorder on X.If X is a preorde red linear space and  X is a partial ord er, then w e say that X is a partially orde re d linear space,andreferto X simply as a v ector order. There is an intim ate relation between convex cones and v ector (pre)or ders. First of all, by using the conve x cones of a linea r space X, we can obtain v arious vector preorders o n X, and thus make X a preordered linear space in a variety of wa ys. To see this, take any nonem pt y subset C of X, and define the bin ary relation  C on X as follows: x  C y if and only if x ∈ y + C. It is easy to see that  C isavectorpreorderonX iff C is a conve x cone — in this case the preorde r  C is called the preorder induced b y C. Conversely, every vector preorder on X arises in this wa y. Indeed , if  is suc h a preorde r, then X + ():= {x ∈ X : x  0} is a convex cone — called the positiv e cone induced b y  —and we have  =  X + () .Moreover, is a p artial order on X iff X + () is poin ted . 6 The upshot is that there is a one-to-one correspondence between convex cones in alinearspaceX and v ector preorders on X. The idea is rather transparent in R. For an y two real n u mbers a and b, the statement a ≥ b is equivalent to a −b ≥ 0, so since a = b +(a−b), another w a y of saying a ≥ b is to say a ∈ b + R + . That is, the natur al order ≥ of R is simply the vector order on R induced by the conv ex cone R + .Put differently, ≥ =  R + . For any n ∈ N, exactly the same situation ensues in R n ,thatis,x ≥ y iff x ∈ y + R n + . O f course, we would have obtained differen t ways of o rdering the real n- v ectors if we used a convex cone in R n distinct from R n + . For instance, if we designated C := {x ∈ R 2 : x 1 > 0 or x 1 =0and x 2 ≥ 0} as our “positive cone” in R 2 , then we w ould be ordering the vectors in R 2 lexicograp hically,thatis, C =  lex (Exa m ple B.1). The smoke must be clearing no w . A preorder  on a linear space X tellsuswhich vectors in X are “positive” in X —avectorx in X is d e em e d “positive” by  iff 6 The following two extreme situations ma y be worth noting here. We have  = X × X iff X + ()=X (thecaseofcompleteindifference), and  = {(x, x):x ∈ X} iff X + ()={0} (the case of full non-compa rability). 324 x  0.Thus,if is a vector preorder, then x  y if and only if x = y + a positive vector in X. In this case, therefore, all there is to kno w about  can be learned from the positive cone induced by . T h is simple observation brings the geometric analys is of c onvex cones to th e fore of o rder-th eoretic inv estiga tions. Notation. As noted above, we denote the preorder associated w ith a g iven pre- ordered linear space X by  X . In turn, the positiv e cone induced by  X is denoted by X + , that is, X + := {x ∈ X : x  X 0}. Inthesamevein,thestrictly positive cone of X is defined as X ++ := {x ∈ X : x  X 0}, where  X istheasymmetricpartof X (Exerc ise A.7). 7 Clearly, if X is a partially ordered linear space, then X ++ = X + \{0}. Warning. The s tric tly positive cone of R n (with respect to the pa r tial order ≥) equ als R n ++ iff n =1. (Wh y ?) E{dpsoh 2. [1] Let X be a preor d ered linear space, and Y a linear subspace of X. A natural way of making Y a preordered linear s pace is to designate X + ∩ Y as Y + . The induced v ector preorder  Y on Y is then  X ∩(Y × Y ). When Y is endowed with this vector preorder, w e sa y that it is a preordered linear subspace of X. [2] For any nonempty set T, recall that the natural partial order ≥ on R T is defined pointw ise, that is, f ≥ g iff f(t) ≥ g(t) for all t ∈ T. ThepositiveconeinR T is thus the class of all nonne gative-valued map s on T : (R T ) + := R T + . Unless otherwise is stated explicitly, all of the function (and hence sequence) spaces that w e consider in this text (e.g. C[0, 1], C 1 [0, 1],  p , 1 ≤ p ≤∞, etc.) a re ordered in t h is can onical way. That is, for any given nonempty set T and linear subspace X of R T ,Xis thought of as a preordered linear subspace of X —wehaveX + := R T + ∩ X. In this case, we denote the induced vector order on X sim ply by ≥ (instead o f  X ). [3] Let X and Y be two preordered linear spaces. A natural way of making the linear product space X × Y a preordered linear space is to endo w it with the product v ector preorder. That is, w e define  X×Y as (x, y)  X×Y (x  ,y  ) if and only if x  X x  and y  Y y  7 Strictly speaking, I’m abusing the terminology here in that X ++ is not a cone, for it does not contain 0. (Of course, X ++ ∪{0} is a convex cone.) 325 for all x, x  ∈ X and y,y  ∈ Y. The n atural positiv e cone of X ×Y is thu s (X ×Y ) + := X + × Y + . Endowedwiththisorderstructure,X × Y is called a preordered linear product of X and Y.  Exercise 14. H Let X be a preordered linear space. Show that X + is a generating cone in X iff, for any x ∈ X, there exists a y ∈ X suc h that y  X x and y  X −x. Exercise 15. (P eressini) Let X be a preordered linear space. (a) Show that if X con tains a v ector x ∗ such that, for all x ∈ X, there exists a λ > 0 with λx ∗  X x, then X + must be generating. (b) Use part (a) to conclude that the positive cones of any of the following partially ordered linear spaces are generating:  p (with 1 ≤ p ≤∞), the linear space c of all con vergent real sequences, B(T ) (for any nonempty set T ). (c)Showthat R ∞ + is a generating cone in R ∞ even though the result stated in part (a)doesnotapplyinthiscase. Let X and Y be t wo preordered linear spaces, ∅ = S ⊆ X, and f ∈ Y S . We say that f is increasing if x  X x  implies f (x)  Y f(x  ) for any x, x  ∈ S. This map is said to be strictly incre as in g if it is in crea sin g and x  X x  implies f(x)  Y f(x  ) for an y x, x  ∈ S. Equivalen tly, f is incre as ing iff f(x) ∈ f(x  )+Y + whenev er x ∈ x  + X + , and strictly increa sing iff it is in c re a sin g and f(x) ∈ f(x  )+Y ++ whenev er x ∈ x  + X ++ for any x, x  ∈ S. It is obvious that these definitions reduce to the usual definitions of increasing and s trictly increasing functions, respectively, in the case w here X is a Euclidean space and Y = R. E{dpsoh 3. [1] For any t wo preordered linear spaces X and Y, alinearoper- ator L ∈ L(X, Y ) is increa sin g iff L(X + ) ⊆ Y + —suchanL i s called a positive linear operator. Similarly, L ∈ L(X, Y ) is strictly incre as in g iff L(X + ) ⊆ Y + and L(X ++ ) ⊆ Y ++ —suchanL is called a strictly positive linear operato r.Ofcourse, when Y = R, we instead talk of positiv e linear functionals a nd stric tly positive linear functionals , respectiv ely. For instance, the zero functional on X is a positiv e linea r function al whic h is not strictly positive. [2] Take any m, n ∈ N,andletA := [a ij ] m×n ∈ R m×n . Recall that L : R n → R m , defined b y L(x):=Ax, isalinearoperator(ExampleF.6). This operator is positiv e iff a ij ≥ 0 for a ll i and j, and strictly positive iff (a i1 , , a in ) > 0 for e a ch i =1, , m. [3] Each of the m aps (x m ) → x 1 , (x m ) → (x 2 ,x 3 , ) and (x m ) → S ∞ 1 2 i x i are positive linear fu nc tion als on  ∞ . The first two of t hes e are not stric tly positive, but the third one is. 326 [...]... < α ≤ 1 The set of all algebraic closure points of S (in X) is called the algebraic closure of S (in X), and is denoted by al-cl X (S) (Note al-cl X (∅) = ∅.) If al-cl X (S) ⊆ S, we say that S is algebraically closed in X Finally, the set al-cl X (S)\al-intX (S) is called the algebraic boundary of S (in X), and is denoted by al-bdX (S) Geometrically speaking, if x is an algebraic closure point of a... first two illustrations in Figure 10.) Another (more geometric) way of putting Proposition 1 is thus the following: Any convex set with a nonempty algebraic interior can be properly separated from any vector outside its algebraic interior by a hyperplane This simple observation, and its many variants, play an important role in geometric functional analysis and its economic applications It is important... S ⊆ al-intX (S), we say that S is algebraically open in X Geometrically speaking, x ∈ al-intX (S) means that one may move linearly from x towards any direction in the linear space X without leaving the set S immediately (See Figure 4 and Exercise 26 below.) While intuitive (is it?), this definition is completely algebraic — it is not based on a distance function — so it makes sense in the setting of... nonempty convex set S with ri(S) = ∅ The second diagram of Figure 8 may help clarify what is going on here Consider a vector like x in that figure This point is expressed by using 0 and e1 which do not constitute a basis for R2 But then we can find a vector, say e2 , which is linearly independent of e1 , and use the line segment through e2 and x to show that x cannot be in the algebraic interior of co({0,... shortly, however, there are strong links between the algebraic interior and the usual interior of a convex set in a variety of interesting cases ∗ ∗ ∗ ∗ FIGURE G. 4 ABOUT HERE ∗ ∗ ∗ ∗ Whether or not a given set is algebraically open depends crucially on relative to which space one asks the question.8 Indeed, if Y is a linear subspace of X, and S ⊆ Y ⊆ X, then al-intY (S) and al-intX (S) may well be quite... is algebraically closed Exercise 32 Show that every linear subspace of a linear space is algebraically closed Conclude that any linear subspace of a Euclidean space is closed 1.6 Finitely Generated Cones In convex analysis one is frequently confronted with the problem of determining whether or not a given conical (or convex) hull of a set is algebraically closed This issue is usually settled within... small relative to ∞ , so the knowledge of the behavior of L on c0 does + not give us enough ammo to settle the extension problem in the affirmative Indeed, there is no member of c0 in the algebraic interior of ∞ (Exercise 24) The following + important result, which goes back to 1948, shows that in cases where the subspace at hand is large enough to intersect the algebraic interior of the positive cone... similar argument can also be used to prove Corollary 2 without invoking the Hahn-Banach Extension Theorem 1 But, please, do not doubt the power of this theorem — choosing a suitable ϕ in this result will give us much more than the trivial applications we looked at so far Read on! 344 The Krein-Rutman Theorem Let X be a preordered linear space, and Y a preordered linear subspace of X such that Y ∩ al-intX... linear space X such that 0 ∈ al-intX (S) The Minkowski functional of S (relative to X) is the real function ϕS on X defined by 1 ϕS (x) := inf λ > 0 : λ x ∈ S Intuitively, you can think of ϕ 1(x) x as the first algebraic boundary point of S (in S X) on the ray that originates from 0 and passes through x ∈ X Figure 9 illustrates the definition geometrically ∗ ∗ ∗ ∗ FIGURE G. 9 ABOUT HERE ∗ ∗ ∗ ∗ The Minkowski... important insight that stems from Example 8 is that, in an infinite dimensional linear space, algebraically open sets, which have nonempty algebraic interiors by necessity, are “really, really large.” You should carry this intuition with you at all times After all, Example 8 shows that even if we have a convex set S so large that aff (S) is actually the entire mother space, we are still not guaranteed the . a λ > 0 with λx ∗  X x, then X + must be generating. (b) Use part (a) to conclude that the positive cones of any of the following partially ordered linear spaces are generating:  p (with. is denoted by al-cl X (S). (Note. al-c l X (∅)=∅.)Ifal-cl X (S) ⊆ S, we say that S is algebr a ically closed in X. F in ally, the set al-cl X (S)al-int X (S) is called the algebraic boundary. observation brings the geometric analys is of c onvex cones to th e fore of o rder-th eoretic inv estiga tions. Notation. As noted above, we denote the preorder associated w ith a g iven pre- ordered