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LOCAL AFFINE SELECTIONS OF CONVEX MULTIFUNCTIONS
SZYMON W ˛ASOWICZA
BSTRACT . It is well known that not every convex multifunction admits anaffine selection. One could ask whether there exists at least local affine selection.The answer is positive in the finite-dimensional case. The main part of this noteconsists of two examples of non-existence of local affine selections of convexmultifunctions defined on certain infinite-dimensional Banach spaces.
1. I
NTRODUCTION
Given two non-void sets X and Y , a map F : X → Y is called a multifunc-tion or a set-valued function . A (single-valued) function f : X → Y is a selection for F , if f ( x ) ∈ F ( x ) for all x ∈ X . There is a plethora of results concern-ing selections of various kinds, with the Micheal Selection Principle concerninglower semi-continuous selections and the Kuratowski–Ryll-Nardzewski SelectionPrinciple concerning measurable selections as probably the most prominent ones.More recent results connected with Michael Selection Principle were establishedby Zippin [12].When X and Y carry a vector-space structure, it is natural to study affine selec-tions or, at least local affine selections for multifunctions F : X → Y , whichare the objective of this note. This topic was investigated (among others) byA. Lazar [6], A. Smajdor and W. Smajdor [10], E. Behrends and K. Nikodem [3],M. Balaj and K. Nikodem [2] and the present author ( cf. [11]).We denote by n( X ) the family of all non-empty subsets of a set X . Now, if X, Y are (real) vector spaces and D ⊂ X is a convex set, then the multifunction F : D → n( Y ) is said to be convex , if(1) tF ( x ) + (1 − t ) F ( y ) ⊂ F (cid:0) tx + (1 − t ) y (cid:1) for any x, y ∈ D and t ∈ [0 , . When the reversed inclusion is stipulated, F is thencalled concave . Of course, the notation A + B and tA is meant in the Minkowskisense, i.e. , A + B = { a + b : a ∈ A, b ∈ B } and tA = { ta : a ∈ A } for any t ∈ R .Observe that a single-valued function f : D → Y is convex (as a multifunction, i.e. , f ( x ) is identified with a singleton { f ( x ) } ) if and only if f is affine , whichmeans that tf ( x ) + (1 − t ) f ( y ) = f (cid:0) tx + (1 − t ) y (cid:1) (cid:0) x, y ∈ D, t ∈ [0 , (cid:1) . It is easy to see that a multifunction F is convex if and only if its graph Gr F = { ( x, y ) : x ∈ D, y ∈ F ( x ) } Date : November 08, 2016.2010
Mathematics Subject Classification.
Key words and phrases. multifunction, selection, convexity, extension of a function, ˇCech–Stonecompactification. is a convex subset of X × Y . Moreover, if F is convex, then F ( x ) is a convexsubset of Y for any x ∈ D . Indeed, if y , y ∈ F ( x ) and t ∈ [0 , , then by (1) weget ty + (1 − t ) y ∈ tF ( x ) + (1 − t ) F ( x ) ⊂ F ( x ) . The condition(2) (cid:16) tF ( x ) + (1 − t ) F ( y ) (cid:17) ∩ F (cid:0) tx + (1 − t ) y (cid:1) = ∅ seems to be the weakest one to guarantee the existence of an affine selection forthe multifunction F . Indeed, if F ( x ) = { f ( x ) } , where f : X → Y is affine, theabove intersection is a singleton (cid:8) f (cid:0) tx + (1 − t ) y (cid:1)(cid:9) .It is proved in [11, Theorem 1] that the multifunction F mapping a real interval I into the family of all compact intervals in R , admits an affine selection if and only ifthe condition (2) is satisfied. In particular, if either F is convex or concave, then F admits an affine selection.One could ask whether a convex multifunction defined on more general domainadmits an affine selection. There is a number of results going in this direction. Oneof the versions of the classical Hahn–Banach Separation Theorem guarantees theexistence of the linear functional separating two convex subsets of a topologicalvector space. It could be easily utilised to prove the existence of a linear (andhence afine) selection of the certain convex multifunction. Since the problem ofextending functions is strongly related to the problem of the existence of selectionsof multifunctions, we notice that Pełczy´nski in his PhD dissertation [9] dealt withlinear versions of the classical Tietze–Urysohn theorem (and extended further theclassical Borsuk–Dugundji theorem).It is worth mentioning that Edwards [5] proved in 1965 the following separationtheorem: Theorem 1.
Let X be a Choquet simplex, f : X → [ −∞ , ∞ ) a convex uppersemicontinuous function and let g : X → ( −∞ , ∞ ] be a concave lower semicon-tinuous function such that f g on X . Then there exists a continuous affinefunction a : X → R such that f a g on X . This result, read in the context of multifunctions, states that the lower semicon-tinuous convex set-valued function defined on a Choquet simplex, whose valuesare compact intervals, admits a continuous affine selection. This multivalued ver-sion of Theorem 1 was extended in 1968 by Lazar ( cf. [6, Theorem 3.1]) to moregeneral codomains.
Theorem 2.
Let ϕ : X → E be a lower semicontinuous affine mapping froma Choquet simplex X to a Fréchet space E that takes non-empty closed values.Then there exists a continuous affine mapping h : X → E such that h ( x ) ∈ ϕ ( x ) for each x ∈ X . In fact, Edwards proved his result in a form of the necessary and sufficient con-dition for X to be a Choquet simplex. This means that if a convex set X is nota simplex, one could find two functions f, g (as considered in Theorem 1), whichcannot be separated by the continuous affine function. Hence, in general, a convexmultifunction defined on a convex subset of a vector space need not to admit the OCAL AFFINE SELECTIONS OF CONVEX MULTIFUNCTIONS 3 affine selection. Let us have a look at the well known example due to Olsen [8](see also Nikodem [7, Remark 1]). Consider the square D = { ( x, y ) ∈ R : | x | + | y | } and the simplex S ⊂ R with vertices ( − , , , (1 , , , (0 , − , , (0 , , .Observe that S is a graph of a convex multifunction F : D → n( R ) (whose valuesare compact intervals) with no affine selection. Nevertheless, locally it is possibleto put a piece of a plane into S . It means that F admits a local affine selection atevery x ∈ Int D . We develop this observation in the next section.A. Smajdor and W. Smajdor proved in [10, Theorem 6] that if F is defined ona cone with the cone-basis in a (real) vector space and F takes the non-empty,closed (and necessarily convex) values in a (real) locally convex space, then F admits an affine selection.2. C ONVEX MULTIFUNCTIONS WITH LOCAL SELECTIONS
Let X be a topological vector space and let D be a non-empty, convex subset of X with non-empty interior. Moreover, let Y be a real vector space. A multifunction F : D → n( Y ) admits a local affine selection at a point x ∈ Int D , if there existan open neighourhood U ⊂ D of x and an affine function f : X → Y such that f ( x ) ∈ F ( x ) for every x ∈ U .The following finite-dimensional version of Edwards’ theorem is quite easy toobtain. Lemma 3.
Let S be an n -dimensional simplex in R n and Y be a (real) vectorspace. Any convex multifunction F : S → n( Y ) admits the affine selection.Proof. Let a , . . . , a n ∈ R n be the vertices of S and let us choose y i ∈ F ( a i ) ( i = 0 , . . . , n ). Then there exists (the unique) affine function f : R n → Y suchthat f ( a i ) = y i ( i = 0 , . . . , n ). Take x ∈ S . Expressing x as a convex combinationof a , . . . , a n (with coefficients λ , . . . , λ n > , λ + . . . + λ n = 1 ) we get f ( x ) = f n X i =0 λ i a i ! = n X i =0 λ i f ( a i ) = n X i =0 λ i y i . Since ( a i , y i ) ∈ Gr F , by convexity of this graph we arrive at (cid:0) x, f ( x ) (cid:1) = n X i =0 λ i a i , n X i =0 λ i y i ! = n X i =0 λ i ( a i , y i ) ∈ Gr F, whence f ( x ) ∈ F ( x ) , x ∈ X and the proof is complete. (cid:3) The above lemma allows us to prove the existence of local affine selections inthe finite-dimensional case.
Theorem 4.
Let D ⊂ R n be a convex set with a non-empty interior and Y bea (real) vector space. Any convex multifunction F : D → n( Y ) admits a localaffine selection at every interior point of D .Proof. Let x ∈ Int D . Then x is the interior point of some n -dimensional sim-plex S ⊂ D . By Lemma 3 there exists the affine function f : R n → Y such that f ( x ) ∈ F ( x ) for any x ∈ S . In particular, f ( x ) ∈ F ( x ) for any x ∈ U = Int S and f is a desired local affine selection of F . (cid:3) S. W ˛ASOWICZ
3. C
ONVEX MULTIFUNCTIONS WITHOUT LOCAL SELECTIONS
In the infinite-dimensional case the problem of the existence of local affine se-lection looks completely different. Namely, there are convex multifunctions withno local affine selection. The following observations are due to Tomasz Kania(Warwick) who has kindly permitted us to include them here.Let X be a closed linear subspace of a Banach space Y . In the light of theHahn–Banach theorem, the multifunction F : X ∗ → Y ∗ given by(3) F ( f ) = { g ∈ Y ∗ : g | X = f and k g k = k f k} ( f ∈ X ∗ ) assumes always non-void values. Certainly, F ( f ) is convex and weak*-compactfor each f ∈ X ∗ . It is easy to prove that F is a convex multifunction. Proposition 5.
Suppose that X is a closed linear subspace of a Banach space Y such that X ∗ does not embed isometrically into Y ∗ . Then F , as defined by (3) ,admits no local affine selection.Proof. Assume contrapositively that there exists an open neighbourhood U of theorigin such that F admits an affine selection ϕ say, when restricted to U . In partic-ular, k ϕ ( g ) k = k g k for all g ∈ U ; thus ϕ is isometric. Denote by T the affine map E ∗ → F ∗ that extends ϕ . As T , T is a linear, isometric embedding of X ∗ into Y ∗ . (cid:3) Remark 6.
The hypotheses of Proposition 5 are easily met when Y = C [0 , .Indeed, by the Banach–Mazur theorem, C [0 , contains isometric copies of allseparable Banach spaces. The dual space of C [0 , is isometric to L ( µ ) for somemeasure µ and this, in turn, prevents many Banach spaces to embed into it ( cf. [1,proof of Proposition 4.3.8]).For instance take X = ℓ . Then X ∗ ∼ = ℓ ∞ which contains isometrically allseparable Banach spaces. In this case it is plain that any isometric copy of ℓ inside of Y = C [0 , meets the hypotheses of Proposition 5. Remark 7.
When the hypotheses of Proposition 5 are met, the mutlifunction is notlower semicontinuous. Indeed, otherwise by Lazar’s theorem (Theorem 2 in thisnote) it would have admitted an affine selection.Let β N denote the ˇCech–Stone compactification of the discrete space of naturalnumbers. Set(4) F ( f ) = { g ∈ C ( β N ) : g | β N \ N = f and k g k = k f k} ( f ∈ C ( β N \ N )) . Then by the Tietze–Urysohn theorem, F ( f ) is non-empty for every f ∈ C ( β N \ N ) .(It is also closed and convex.) Certainly the multifunction F is convex. Proposition 8.
The multifunction F given by (4) does not admit a local affineselection.Proof. Arguing as in the proof of Proposition 5, we would get a linear embeddingof C ( β N \ N ) into C ( β N ) , however it is known that no such operator exists. (Forexample, the former space does not have a strictly convex renorming but the latterdoes; the possibility of finding a strictly convex renorming passes to subspaces, cf. [4]; here we use the fact that C ( β N \ N ) is isometric to ℓ ∞ /c .) (cid:3) OCAL AFFINE SELECTIONS OF CONVEX MULTIFUNCTIONS 5 R EFERENCES [1] Fernando Albiac and Nigel J. Kalton.
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GraduateTexts in Mathematics . Springer, New York, 2006.[2] Mircea Balaj and Kazimierz Nikodem. Remarks on Bárány’s theorem and affine selections.
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Studia Math. , 116(1):43–48, 1995.[4] Jean Bourgain. l ∞ /c has no equivalent strictly convex norm. Proc. Amer. Math. Soc. ,78(2):225–226, 1980.[5] David Albert Edwards. Séparation des fonctions réelles définies sur un simplexe de Choquet.
C. R. Acad. Sci. Paris , 261:2798–2800, 1965.[6] Aldo J. Lazar. Spaces of affine continuous functions on simplexes.
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Acta Univ. Car-olin. Math. Phys. , 30(2):125–129, 1989. 17th Winter School on Abstract Analysis (Srní, 1989).[8] Gunnar Hans Olsen. On simplices and the Poulsen simplex. In
Functional analysis: surveys andrecent results, II (Proc. Second Conf. Functional Anal., Univ. Paderborn, Paderborn, 1979) ,volume 68 of
Notas Mat. , pages 31–52. North-Holland, Amsterdam-New York, 1980.[9] Aleksander Pełczy´nski. Linear extensions, linear averagings, and their applications to lineartopological classification of spaces of continuous functions.
Dissertationes Math. RozprawyMat. , 58:92, 1968.[10] Andrzej Smajdor and Wilhelmina Smajdor. Affine selections of convex set-valued functions.
Aequationes Math. , 51(1-2):12–20, 1996.[11] Szymon W ˛asowicz. On affine selections of set–valued functions.
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