JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 204, 522᎐544Ž. 1996 ARTICLE NO. 0452
Complex Strongly Extreme Points in Quasi-Normed Spaces*
Zhibao Hu
Miami Uni¨ersity, Oxford, Ohio 45056-16411
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Douglas Mupasiri
Uni¨ersity of Northern Iowa, Cedar Falls, Iowa 50614-0506
Submitted by Richard M. Aron
Received April 13, 1994
We study the complex strongly extreme points ofŽ. bounded subsets of continu- ously quasi-normed vector spaces X over ރ. When X is a complex normed linear space, these points are the complex analogues of the familiarŽ. real strongly extreme points. We show that if X is a complex Banach space then the complex
strongly extreme points of BX admit several equivalent formulations some of which are in terms of ‘‘pointwise’’ versions of well known moduli of complex convexity. We use this result to obtain a characterization of the complex extreme points of
B Ž . and B Ž . where 0 - p - ϱ, X and each X , j I, are complex l pjjX gIpL , X j g Banach spaces. ᮊ 1996 Academic Press, Inc.
1. INTRODUCTION
The study of complex geometry dates back to a 1969 paper of E. Thorp and R. Whitleywx 14 . In that paper the authors characterized the complex
* The results in this paper constitute part of a Ph.D. thesis submitted by the second author in 1992 at Northern Illinois University, Dekalb, Illinois. The authors thank Professors Patrick N. Dowling and Mark A. Smith for their advice and helpful comments.
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0022-247Xr96 $18.00 Copyright ᮊ 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. COMPLEX STRONGLY EXTREME POINTS 523
Banach spaces X for which the strong maximum modulus theorem holds for X-valued analytic functions defined on an open connected subset of the complex plane in terms of a geometric condition called ‘‘complex strict convexity.’’ In 1975 J. Globevnikwx 3 brought complex geometry into sharper focus by introducing complex uniform convexity, a notion which is stronger than complex strict convexity, as a natural generalization ofŽ. real uniform convexity. Globevnik then proceeded to show that for a measure space
Ž.⍀,⌺,, the complex space L1 Ž., X is not only complex strictly convex as Thorp and Whitley had shown inwx 14 , but is in fact complex uniformly convex. Almost a decade later, a fundamental paper on complex convexity of quasi-normed linear spaces was published by W. J. Davis, D. J. H. Garling, and N. Tomczak-Jaegermannwx 1 . The paper considered a number of topics related to complex convexity. The topics discussed included various moduli of complex convexityŽ which give a measure of the plurisub- harmonicity of the quasi-norm unlike theŽ. real moduli of convexity which give a measure of the convexity of the unit ball of normed spaces. ; the connection between the behavior of suitable martingales taking values in complex quasi-normed spaces and the existence of equivalent complex uniformly convex quasi-norms; the relation of complex convexity with cotype; and results concerning complex Banach lattices. In this paper we study the complex strongly extreme points of the closed unit ball of continuously quasi-normed spaces X over ރ. When X is a complex normed linear space, these points are the complex analogues of the familiarŽ. real strongly extreme points. We show that if X is a complex Banach space then the complex strongly extreme points of B admit several equivalent formulations some of which are in terms of ‘‘pointwise’’ ver- sions of well known moduli of complex convexity. We use this result to obtain a characterization of the complex extreme points of B Ž . and l pjjX gI B where 0 - p - ϱ, X and each X , j I, are complex Banach L pŽ , X . j g space. We remark that in the real case it is well known that a norm one function f in LpLŽ., X is a strongly extreme point of B Ž,X.if and only Ž. Ž. if f r5f 5is a strongly extreme point of BX for -almost all in the support of f. The survey articlewx 12 gives a more detailed account of this and related issues. Our results for the case of the complex function spaces
LpŽ.,Xparallel the known results in the real case. The paper is organized as follows. In Section 2 we review some basic definitions and relevant preliminaries. In Section 3 we characterize the complex strongly extreme points of BX . In Section 4 we characterize the complex strongly extreme points of B and B for 0 - p - ϱ. L ppŽ , X . l Ž X . Our arguments for the complex space LpŽ., X are direct and straightfor- ward and apply to the real case as well. 524 HU AND MUPASIRI
2. BASIC DEFINITIONS AND PRELIMINARIES
Throughout this paper all vector spaces are assumed to be over the field of complex numbers.
DEFINITION 2.1. Let X be a vector space. A map 55и : X ª w0, ϱ.is called a quasi-seminorm if
Ž.155<<55␣xs␣x᭙␣gC,xgX Ž.2 there exists a constant C s CX Ž ,55.G1 such that
5xqy 5FCxŽ. 55q 55y for all pairs x, y g X.
We call 55и a p-seminorm where 0 - p F 1 if in addition p p p Ž.3 55xqy F55x q55y ,x,ygX. If also 55x s 0 implies x s 0, then 55и is a quasi-norm Žrespectively, p-norm. if conditionsŽ.Ž.Ž 1 ᎐ 2 respectively, conditionsŽ.Ž.. 1 ᎐ 3 hold. The pair Ž X,55. , or simply X if there is no ambiguity, is then called a quasi-seminormed, p-seminormed, quasi-normed, p-normed,ornormed space according to whether 55и is a quasi-seminorm, a p-seminorm, a quasi-norm, a p-norm, or a norm, respectively. We remark that 55и induces a metrizable vector topology on X. If this topology is complete, then the pair Ž X,55. is called a quasi-Banach space. In case 55и is a quasi-norm which is uniformly continuous on 55и -bounded subsets of X, then the pair Ž X,55. or simply X, if no confusion results, is called a continuously quasi-normed space. Note that by propertyŽ. 3 of Definition 2.1, a p-seminorm is uniformly continuous with respect to the uniformity it generates on X. We also note that according to the Aoki᎐Rolewicz theoremwx 9 , every quasi-norm is equivalent to a p-norm for some p,0-pF1. Consequently, it is customary to assume that all quasi-norms are p-norms for some p. We shall not need to make this assumption here.
DEFINITION 2.2. Let X be a quasi-normed space. An upper-semicon- tinuous function : X ª wyϱ, ϱ. is said to be plurisubharmonic if for every x, y g X
d 2 i Ž.x F H Ž.x q ey . 0 2 COMPLEX STRONGLY EXTREME POINTS 525
DEFINITION 2.3. Davis, Garling, and Tomczak-Jaegermann call a con- tinuously quasi-normed space Ž X, 55.locally PL-con¨ex if for any x, y g X, there exists ␦ s ␦Ž.x, y ) 0 such that
d 2 i 55xFH 5xqre y 5 for all 0 - r - ␦ . 0 2
Remark. By Proposition 2.2 ofwx 1 , the defining condition for PL-con- vexity is equivalent to requiring that log55и be plurisubharmonic. Since, by Jensen’s inequality, 55и is plurisubharmonic if log 55и is plurisubharmonic, a continuously quasi-normed space is locally PL-convex if and only if it is equipped with a plurisubharmonic normŽ cf. Lemma 2.2 ofwx 8. . It thus follows that in the setting of continuously quasi-normed spaces the terms locally PL-con¨ex and PL-con¨ex are equivalent. A superclass of the PL-convex spaces is the class of ‘‘ A-convex’’ spaces introduced by N. Kalton inwx 8 . Kalton calls a complex quasi-normed space A-con¨ex if it admits an equivalent plurisubharmonic norm. These spaces have the characteristic ability to control the behavior of vector-valued analytic functions taking values in them in the following way. Let X be an
A-convex space and let AX0Ž.denote the collection of all X-valued functions which are continuous on the closed unit disc of the complex plane and analytic in its interior. X is said to satisfy theŽ. weak Maximum Modulus PrincipleŽ. MMP if there is a constant M ) 0 so that for all Ž. fgAX0 we have
f Ž.0 F M max fzŽ.. < A result of Kaltonwx 8 says that the A-convex quasi-Banach spaces are precisely the spaces X satisfying the MMP. We conclude this section by recalling the definitions of two continuously quasi-normed quasi-Banach spaces whose complex geometry will be of particular interest to us in Section 4. EFINITION ϱ Ž. D 2.4. Let 0 - p - and let XjjgIbe a family of complex Ž. Banach spaces. The l pi-sum of the family X igIis the linear space Ž. Ä <Ž. 55 ϱ4 lXpjjgIj[f:IªDgIjXfjgX jfor all j and f p - where the 55и 55 ŽÝ 5Ž.5p.1rp quasi-norm on X is given by f p s jg I fj and Ý 5 Ž.5p ÄÝ 5Ž.5p 4 jg Ijfj [sup gFfj :F;Iand F is finite . DEFINITION 2.5. Let X be a complex Banach space. For 0 - p F ϱ,we denote by LpŽ.⍀, ⌺, ; X the Lebesgue᎐Bochner space of -equivalence 526 HU AND MUPASIRI classes of strongly measurable functions f: ⍀ ª X for which Ž.p Ž. Ä Ž. 4 H⍀ 5 f5d -ϱif 0 - p - ϱ and ess sup 5f 5: g ⍀ - ϱ if p s ϱ, quasi-normed by 1rp p ¡ Hf Ž. d Ž. if 0 - p - ϱ 55fps~ž/⍀ ¢ess supÄ4f Ž. : g ⍀ if p s ϱ. For convenience we shall use the abbreviated notation LpŽ., X for LpŽ.⍀,⌺,;X. 3. A CHARACTERIZATION OF THE COMPLEX STRONGLY EXTREME POINTS OF BX In the proof of the characterization of the complex strongly extreme points of the closed unit ball of a complex Banach space below, we shall need a special case of an observation made by S. J. Dilworth in the proof ofŽ. a implies Ž. d in Proposition 2.2 ofwx 2 .Ž A proof of the observation was not given inwx 2 but, in the opinion of the author, the proof may not be entirely obvious.. Followingwx 10 , we record the observation as a lemma and state it in a form which is more suitable for our purposes. A detailed proof of the lemma can be found inwx 10 . LEMMA 3.1. Let X be a complex Banach space,0-p-ϱ,and 0 -  - i 1r2. Let x, y g X satisfy55 x s 1, 55y F 1, and 5 x q ey5)1qfor some ,0FF2.Suppose x* g X* is a norming functional of x and x*Ž.ys␣.Then Ž.1 if <<␣ F r6, we ha¨e  ii 55< Ž. Ž . 2 for any complex number , there exists a real number t00s tp, 0-t0 -1r2such that p i ip <<1qeq) <<1qeqt0 ž/33 for all ,0FF2. DEFINITION 3.2. Let ŽX,55. be a continuously quasi-normed complex space. Let K be aŽ. not necessarily convex bounded subset of X. We say a COMPLEX STRONGLY EXTREME POINTS 527 point x g K is a complex strongly extreme point of K if and only if for each ⑀)0 there is a ␦ s ␦⑀Ž.)0 such that x is the center of no one-dimen- Ž. sional disc which is contained in K q B␦ 0 and has diameter at least 2⑀; Ä 4 Ž. i.e., whenever x q zy: < Note.1IfŽ.xgc-str. ext K then x g c-ext K. Žx g c-ext K if Äx q zy: < X i HϱŽ.⑀sinfÄ4 supÄ45 x q ey55:0FF2 y1: x5s 1, 5y5s ⑀ . AlsoŽ seewx 1. a continuously quasi-normed complex space X is said to be X Ž. uniformly PL-con¨ex if and only if Hp ⑀ ) 0 for any, and hence all p, X 0-p-ϱwhere the modulus Hp Ž.⑀ is defined by 1rp d Xi2 p HpŽ.⑀sinf H55x q ey y1: 5555x s 1, y s ⑀ . ½5ž/0 2 Observe that if X is a PL-convex quasi-Banach space, then either by applying the Maximum Modulus Principle to the polynomial z ¬ x q zy for fixed, x, y g X or bywx 11, Theorem 17.5, p. 337 , we may, without loss of X X generality, take the infimum in the definitions of Hp Ž.⑀ and Hϱ Ž.⑀ over all y g X with 55y G ⑀. Fix x g X with 55x s 1. The following moduli of convexity are derived XX X from c , Hϱ , and Hp Ž.0 - p - ϱ , respectively: X cŽ.x,␦[supÄ455y : y g X, 5x q zy 5- 1 q ␦ ᭙z g Bރ, X i HxϱŽ.,⑀[infÄ4 supÄ45x q ey55:0FF2 y1: y5G ⑀ , and 1rp d Xi2 p HxpŽ.,⑀[inf H55x q ey y1: 55y G ⑀ . ½5ž/0 2 528 HU AND MUPASIRI X Ž. XŽ. It is clear that for each x g X, Hxϱ ,⑀GHxp ,⑀for any 0 - p - ϱ. We shall use this fact in the proof of the sufficiency of the conditions in Theorem 4.2. The next theorem shows that each of these moduli of convexity can be used to characterize the complex strongly extreme points of BX if X is a complex Banach space. THEOREM 3.3. Let X be a complex Banach space. Let x g SX . Then the following statements are equi¨alent: Ž. 1 xgc-str. ext BX ; Ž. Ž . 2If ynn is a sequence in X such that5555 x q y ª 1, x y ynª 1, 55xqiynnnª 1 and 55 x y iy ª 1, then y ª 0; X Ž.3 lim q Ž.x, ␦ 0; ␦ ª 0 c s Ž. ÄÄ 4 4 Ž XŽ. 4᭙⑀)0, inf sup 55< 2s552xs 5xqzynnq x y zy 5 F5555xqzynnq x y zy 11 -1yq1qs2 nn if either 55x q zynn- 1 y 1rn or 55x y zy - 1 y 1rn for some z g Bރ,a contradiction. Thus for all n g ގ and for all z g Bރ , we have 11 1yF55x"zyn F 1 q . nn COMPLEX STRONGLY EXTREME POINTS 529 Consequently, 55555x q ynnª 1, x y y ª 1, x q iy n 5ª 1, and 5z y iy n 5 ª1. But 55ynnG ⑀ for all n,so y ¢0. Ž.3« Ž.4 . Suppose Ž. 4 fails. Then there exists ⑀ ) 0 such that for each ␦)0 there is yŽ.␦ in X with 5yŽ.␦ 5G ⑀ such that supÄ5x q zyŽ.␦ 5: < i i i 555555xqeysx*иxqeyGx*Ž.xqey i )<1qe< for all ,0FF2. We now consider two cases according to the size of 55yrelative to 55x . Case 1. 55y F 55x s 1. Then 3 dt 2ip 0 H 55xqey Gmin 1 q ,1q⑀0 , 2½554 0 Ž. Ž. where t00s tpand ⑀ 0s ⑀ 0p are chosen so that 0 - t0- 1r2 and Ä2 ipŽ. 4 0-⑀00-1r2. Hence inf H 5x q ey5 dr2:55yG⑀)1. 530 HU AND MUPASIRI Case 2. 55y ) 55x s 1. Choose x* g X* such that 5x* 5s 1 and x*Ž.y s55y, and let s x*Ž.x . Then << F 1. So we have i i 5 yqex5555sx*иyqex5 i Gx*Ž.yqex i s55yqe i s55yи1qe 55y i )1qe0 Ž.55y)1, where 0 s r55y , holds for all g wx0, 2 . Now arguing as in Case 1, we get that d d 2 ipp2 i HH55xqey s 55yqex 002 2 3 t0 Gmin 1 q ,1q⑀ , ½554 Ž. Ž. where 0 - t00s tp-1r2 and 0 - ⑀0s ⑀ 0p - 1r2. Remark. It is easy to prove that under the hypotheses of Theorem 3.3 each of the following statements is equivalent to any one, and hence all, of the statements in Theorem 3.3: Ž.1 ᭙⑀)0, infÄÄ sup 55< 4. A CHARACTERIZATION OF THE COMPLEX STRONGLY EXTREME POINTS OF B AND B L ppŽ , X . l Ž X j. FOR 0 - p - ϱ We begin this section with a characterization of the complex strongly Ž. extreme points of the closed unit ball of the space lXpjjgIwhere ϱ Ž. 0-p-and XjjgIis a family of complex Banach spaces. We will use this result to reduce the characterization of the complex strongly extreme points of B , for 0 - p - ϱ and X a complex Banach space, to the L pŽ , X . COMPLEX STRONGLY EXTREME POINTS 531 case where Ž.⍀, ⌺, is a finite measure space. In the sequel we shall denote the support of a function f: X ª Y by sptŽ.f , i.e., spt Ž.f [ Äxx< gXand fxŽ./0.4 HEOREM ϱ Ž. T 4.1. Let 0 - p - and let XjjgI be a family of complex Ž. Banach spaces. Put X s lXpjjgIX.Then f g c-str. ext B if and only if 55f 1and fŽ. j 5fjŽ.5 c-str. ext B for all j in the support of f. p s r g X j Proof. The proof of the necessity of the conditions is straightforward and will therefore be omitted. To prove the sufficiency of the conditions, let ⑀ ) 0. Choose a finite Ä4Ž. subset I01s i ,...,in;spt f such that p p⑀ Ý fjŽ. - if 0 - p F 1 4 jgI_I0 or 1rp p ⑀ Ý fjŽ. - if 1 - p - ϱ. ž/4 jgI_I0 Let ⑀ if 0 - p 1 ¡ 1rp F Ž.4n и maxÄ4fj Ž.:jgI0 ⑀1s~ 1p Ž.3y2r ⑀ if 1 - p - ϱ. 1r p ¢4n и maxÄ4fjŽ.:jgI0 For each j g I0 , choose ␦jj) 0 such that whenever x g Xjand 55x jG ⑀1, p fjŽ. d 2 i H qexjj)1q␦.1Ž. 0 fjŽ. 2 Ä p Ž.p4ÄŽ.p 4 Let ␦ s min ␦1,...,␦n, ⑀ r4, ⑀r4 , where s min 5fj5:jgI0. Then ␦ ) 0. Let g g X satisfy 55g G ⑀. Case 1. 0 - p 1 and Ý 5 gjŽ.5p Ž.34⑀p. Then F jg I _ I0 G r p d 2 i H f q eg 0 2 ppd d 2i2i sÝÝHHfjŽ.qegj Ž. q fjŽ.qegj Ž. 002 2 jgIj00gI_I 532 HU AND MUPASIRI ppp2 d GÝÝÝfjŽ. qHgjŽ. y fjŽ. ž/2 0 jgIj000gI_IjgI_I 2 pp GÝ fjŽ. q ⑀ 4 jgI0 p pp⑀ GÝÝfjŽ. q fjŽ. q 4 jgIj00gI_I ⑀p s1q 4 G1q␦, where the first inequality follows from the plurisubharmonicity of the norm of each of the Banach spaces Xj, and from the fact that being a p-norm, 55и psatisfies the triangle inequality. This proves Case 1. Case 2. 0 - p 1 and Ý 5 gjŽ.5p-Ž.34⑀p. Then since 55g ⑀, F jg I _ I0 r G we must have that Ý 5 gjŽ.5p ⑀p 4. So there exists j I such that jg I0 G r 00g Ž.p p Ž. Ž.1rp 5gj005G⑀r4n. Thus 5gj 5G⑀r4n so gjŽ.00gjŽ. G fjŽ.00maxÄ4fj Ž.:jgI 1rp ⑀rŽ.4n G maxÄ4fjŽ.:jgI0 s⑀1. Consequently, ppd d 2i2i HHfqeg s fjŽ.00qegj Ž. 002 2 pd 2i qÝH fjŽ.qegj Ž. 0 2 jgI_Ä4j0 Ž.2 1␦55fjŽ.p 55fjŽ.p GŽ.qj0 0q Ý jgI_Ä4j0 p 1 ␦ fjŽ. s q j0 0 G1q␦ Ž.by definition of ␦ . COMPLEX STRONGLY EXTREME POINTS 533 The inequalityŽ. 2 holds because of Ž. 1 and the fact that the norm of X is j0 plurisubharmonic. This proves Case 2. Case 3. 1 - p - ϱ and ŽÝ 5 gjŽ.5p.1rp Ž1 21rp.Ž⑀4 . . Then jg I _ I0 G q r d 2 i p H55fqeg 0 2 ppd d 2i2i sÝÝHHfjŽ.qegj Ž. q fjŽ.qegj Ž. 002 2 jgIj00gI_I p GÝ fjŽ. jgI0 p 1rp 1rp 2 ppd qHÝÝgjŽ. y fjŽ. ž/ž/2 0 jgI_Ij00gI_I p ⑀⑀d p 2 1p GÝ fjŽ.qH Ž1q2r . y 0 442 jgI0 p p ⑀ sÝ fjŽ. q2 ž/4 jgI0 p pp⑀ GÝÝfjŽ. q fjŽ. q ž/4 jgIj00gI_I p p⑀ sÝfjŽ. q ž/4 jgI G1q␦. This proves Case 3. Case 4. 1 - p - ϱ and ŽÝ 5 gjŽ.5p.1rp-Ž1 21rp.Ž⑀4 . . Then jg I _ I0 q r 1rp p ⑀F55gsž/ÝgjŽ. jgI 1rp 1rp pp Fž/ž/ÝÝgjŽ. q gjŽ. jgIj00gI_I 1rp⑀ p1p -Ý gjŽ.q Ž1q2.r . ž/ 4 jgI0 534 HU AND MUPASIRI ThusŽŽ 3 21r p..4 и ⑀ ⑀ Ž1 21rp.Žи ⑀ 4 .- ŽÝ 5gjŽ.5p.1rp and y r s y q r jg I0 soŽŽŽ 3 21r p...4 ⑀ p- Ý 5gjŽ.5p. Hence there exists j I such that y r jg I0 g 0 Ž. ŽŽ 1r p. 1r p. 5 gj0 5) 3y2 r4n ⑀and so 1r p gjŽ.0 Ž3y2 .и⑀ )\⑀. 1rp 1 fjŽ.0 4n maxÄ4fjŽ.:jgI0 Consequently, d p d 2 i p 2 i HH55fqeg s fjŽ.00qegj Ž. 002 2 pd 2i qÝH fjŽ.qegj Ž. 0 2 jgI_Ä4j0 pp 1␦fjŽ. fjŽ. GŽ.qj0 0q Ý jgI_Ä4j0 pp fjŽ.␦ fj Ž . sÝ qj0 0 jgI G1q␦ Ž.by Eq.Ž. 2 . This proves Case 4, and hence completes the proof of the theorem. Remark. We have actually proved more than we asserted in the state- ment of the preceding theorem. The proof shows that the theorem remains Ž. valid if we replace X in the statement of the theorem by Y s lYpjwhere each Žyj, 55и .is a quasi-normed space such that the quasi-norm 55и j is p p p p-subadditive, that is, 55x q y jjjF 55x q 55y for every pair of vectors x,ygYj. We now prove a similar result for the LpŽ., X spaces. THEOREM 4.2. Suppose 0 - p - ϱ, Ž.⍀, ⌺, is a complete, positi¨e Ž. measure space, X is a complex Banach space, and f g Lp , X . Then f c-str.ext B if and only if55 f 1 and fŽ. 5fŽ. 5 c- g L pŽ , X . p s r g Ž. str.ext BX for -almost all g spt f . Proof. Ž.Sufficiency . Since f has -finite support we may, by Theorem 4.1, assume that Ž.⍀, ⌺, is a finite measure space. COMPLEX STRONGLY EXTREME POINTS 535 Observe that f c-str.ext B if and only if HfX Ž.,⑀)0 for all g L pŽ , X . ϱ X ⑀)0 where Hfϱ Ž.,⑀is the modulus defined by X ⑀ 55 HfϱŽ.,[inf½5 sup f q zg py 1 55g ⑀ pG < ŽŽsee Definition 3.2 and the proof of 4.« Ž3. in Theorem 3.3. . Let ⑀ ) 0. X We shall show that Hfϱ Ž.,⑀)0 by proving than the much stronger condition p d 2 i inf H f q egp )1 55gpG⑀ 0 2 Ž. holds. To this end, let g g Lp , X and 55g p G ⑀. The proof will be 2 i pŽ. complete once we show that H0 5 f q eg5 p dr2G1qwhere the Ž. Ž. constant  s  f, p, ⑀ ) 0 is independent of g. Since Lp , X ( Ž Ž. . Ž Ž. . Ž. Lpp<⍀_spt f , X [ L p i HfŽ.qeg Ž. d Ž. Eg ppp G2yHHgŽ.d Ž.y f Ž. d Ž. EEgg ppp s2yHHgŽ.d Ž.y g Ž. d Ž. ž/⍀⍀E _0 p yHfŽ.d Ž. Eg 536 HU AND MUPASIRI ⑀⑀p pp G2y⑀yHž/ž/иfŽ. d Ž.y ž/⍀_Egkk Ž.since Ž.Eg- ␣ ⑀⑀p pp p G2y⑀y HfŽ.d Ž.y ž/kk⍀ ž/^`_ 1 ⑀⑀p s pp s2y⑀yy ž/ž/kk ⑀ )2 ,Ž. by our choice of k k ⑀⑀ sq kk p ⑀ )HfŽ.d Ž.q Eg k for each g wx0, 2 . So, we have, by Fubini’s Theorem, that ppd d 2i2i HHHfqegp s fŽ.qeg Ž. d Ž. 002⍀ 2 pd 2i sHH fŽ.qeg Ž. d Ž. 0 ⍀_Eg 2 p d 2 i qHH fŽ. qeg Ž. d Ž. 0 Eg 2 pd 2 i )HHfŽ. qeg Ž. d Ž. ⍀_Eg0 2 2 p ⑀ d qHHfŽ. d Ž.q 0ž/E k2 g pp⑀ GHHfŽ.d Ž.q f Ž. d Ž.q ⍀_EEggk ⑀ s1q. k COMPLEX STRONGLY EXTREME POINTS 537 Since the constant ⑀rk is independent of the function g, this proves the theorem for Case 1. Ž. X qq Case 2. EgpG ␣. Let ␦ : SX= ޒ ª ޒ be the map defined by d 2ip Ž.x,⑀¬inf H55x q ey y1. 55yG⑀ 0 2 X Note that for each fixed ⑀ ) 0, ␦pŽ.x, ⑀ is an upper semicontinuous q function from SX to ޒ and hence is Borel measurable. To see this define q 2 for each y g X with 55y G ⑀, a function f yX: S ª ޒ by x ¬ H05x q ip ey5Ž. dr2. Then by the Lebesgue Dominated Convergence Theorem, Ž. we have for every sequence xnnin X with x ª x g X, d 2ip lim fxynŽ.slim H55xnq ey nªϱ xªϱ0 2 d 2 ip sHlim 55xnq ey 0 nªϱ 2 d 2 i p sH55xqey 0 2 sfxyŽ.. ␦ XŽ.и ⑀ ÄŽ.и Hence f ypis continuous. Consequently the map , s inf 5y5G⑀f y: 4 Ž 55yG⑀is an upper semicontinuous function on SX seew 6, Theorem 7.22, X p. 89x. . Furthermore, for fixed ⑀ ) 0, the composite function ␦pŽŽ.h и , ⑀ .is Ž X Ž. -measurable for every -measurable function h: ⍀ ª SXpbecause ␦ и, ⑀ q is a Borel measurable function on SX = ޒ .. Ž. Ž. Ž. We may assume that f r5f 5g c-str.ext BX for all g spt f . Put ⑀ X 11 Anp[g⍀:␦fŽ.rf Ž.,)and f Ž. ) . ½5ž/kn n Then by the preceding remarks each An is -measurable. Furthermore, иии иии ϱ Ž. A12;A; ; Ann; A q1; and D ns1Ans spt f because fŽ. 5fŽ.5 c-str.ext B for all sptŽ.f . Thus lim ŽA . r g Xng ªϱns ŽϱA.ŽŽ..spt f . Choose n ގ such that Ž.ŽŽ..A ) spt f Dns1 n s 0 g n0 538 HU AND MUPASIRI y ␣r2. Then A E E E Ac Ž.ng00l s Ž. gy Ž. gnl ␣AcsptŽ.f GyŽ.n0l ␣ )␣y 2 ␣ s. 2 Ž. The first inequality holds since Eg ; spt f and the second inequality holds because A sptŽ.f so that spt Ž.f A Žspt Ž.f Ac.and nn00; s j l n0 hence ŽŽ.spt f Ac.ŽŽ..Ž. spt f A since is a finite measure. l nn00s y For each A E , we have g ng0l p fŽ.⑀ fŽ. d 1 X 2 i ␦p , s inf H q ey y1) ž/fŽ.kf55yG⑀rk0 Ž.2n0 so, p f Ž. g Ž. d 1 2 i H qe y1) 0 fŽ. f Ž. 2 n0 gŽ.⑀ G , definition of Eg , ž/fŽ.k hence 1 p d 1 2 i pHf Ž. q eg Ž. )1q fŽ. 0 2 n0 or equivalently, p d 1 2 i p Hf Ž. q eg Ž. )f Ž. 1q . 0 2 ž/n0 Therefore by Fubini’s theorem, ppd d 2i2i HHHfqegp s fŽ.qeg Ž. 002⍀ 2 pd 2i sHH fŽ.qeg Ž. d Ž. 0 A E 2 ng0l COMPLEX STRONGLY EXTREME POINTS 539 p d 2 i q HH f Ž. q eg Ž. d Ž. 0⍀Ž.AE 2 _ng0l pd 2 i sHHfŽ.qeg Ž. d Ž. A E0 2 ng0l pd 2 i qHHfŽ.qeg Ž. d Ž. ⍀Ž.AE0 2 _ng0l p 1 GH fŽ.1qdŽ. AngE ž/n0 0l p qH fŽ.d Ž. ⍀Ž.AE _ng0l pp1 sHHfŽ.d Ž.q f Ž. d Ž. ⍀ AngE ž/n0 0l p 11 G1qH dŽ. AngEž/ž/nn00 0l definition of A Ž.n0 1pq1 1AE sqŽ.ng0l ž/n0 1 pq1 ␣␣ G1qsince Ž.Angl E ) . ž/ ž0 / ž/n022 This establishes Case 2 and hence completes the proof of the sufficiency of the conditions. Ž.Necessity It is clear that 55f p s 1. For n G 1 let 1 Ans x g X : for all m G 1 there exists ymŽ.with ym Ž.G ½ n 1 and sup x q zyŽ. m - 1 q . m5 < ϱ LAIM C . Each Ann is a Borel set and D s1AnXs S _ c-str.ext B X. 540 HU AND MUPASIRI Proof of Claim. Fix m G 1 and put 1 An,ms x g X : there exists ymŽ.,x gXwith ym Ž.,x G ½ n 1 and sup x q zyŽ. m, x - 1 q . m5 < Then each An, m is an open set. To see this, let x0 g An, m. Choose Ž. 5Ž55Ž.5 ym,x00gXso that ym,x G1rnand sup xqzyŽ. m, x00F x q zy Ž. m, x 0q 55x y x 0 -rq⑀. 5Ž.55⑀ Ž.5 So sup 1 f Ž. Ek,msg⍀: FfŽ. Fmand g Ak ½5mfŽ. for each pair of positive integers k and m. Then Ž.Ek, m - ϱ for Ž. Ä ⍀ each pair k and m and DDm G 1 k G 1 Ek, m sspt f l g : ŽŽ .5Ž.5. 4 Ž. frfgSXX_c-str.ext B . Hence DD mG1kG1Ek,m)0by hypothesis. So there exists positive integers k and m such that Ž.E 00 k00,m )0. Let ⑀ Ž.ŽŽ..1 km E 1rp. Since f c-str.ext B there s r 00 k00,mLg pŽ,X. ␦ ␦⑀Ž. Ž. Ä5 exists s )0 such that whenever g g Lp , X and sup< z < F1 f q zg 5< Ž. BnmlBsлif n / m, each Bng ⌺ and satisfies diam fBn-2, and E ϱB. For each n ގ choose B and set f f и k 00, mnsDs1nnng g 1s ϱ ⍀ EnÝ1fŽ.nB. Then _ k00,mnq s ϱ p 55ffp fŽ . Ž. f Ž. Ž. d Ž. 1ypsH Ý nBnny B E k00,mns1 ϱ p fŽ . Ž. f Ž. Ž. d Ž. sH Ý nBnny B E k00,mns1 ϱ p sÝHfŽ.nyf Ž.d Ž. Žby Fubini’s Theorem. B ns1 n ϱ p FÝŽ.2Hd Ž.Ž.diam fB Ž.n-2 B ns1n p Ž.2E . sŽ.k00,m Therefore 1 p 55ff2E r 1ypFŽ.Ž.k00,m ␦ 1p 2 иE r s 1pŽ.Ž.k00,m 6E r Ž.Ž.k00,m ␦ s. 3 Ž Also 55f11pppF 5f y f 5q 55f F 1 q ␦r3. Choose ␦ 1) 0 such that 1 q ␦ .Ž1 ␦ 3 . - 1 ␦ 2. Since B E for each n, it follows 1 q r q r nnkg ; 00,m from the definitions of A and E that there exists x X such that kk00,mn0 g 55 Ä5Ž. 5Ž. 55<<4 ␦ xn G1rk0 and sup< z < F1 f nnnr f q zx : z F 1 F 1 q 1. Let g Ýϱ5fŽ.5x. Then sns1 nnBn p ϱ p 1r 55g fŽ .x Ž.d Ž. psHÝ nnBn ž/⍀n1 s 1p ϱ pr fŽ .x Ž.d Ž. sH Ý nnBn E ž/k00,mn1 s ϱ 1rp p p sÝHfŽ.nnxd Ž. ž/B ns1n 542 HU AND MUPASIRI p 1 p ϱ 1 r G Ý HdŽ. ž/mk00 Bn ž/n1 s p 1rp 111p EB Er sŽ.k00,mksŽ.Ž.00,m ž/ž/mk00 mk00 s⑀ Ž.see definition of ⑀ . Moreover, for all g ⍀ and for all z with < f1Ž. q zg Ž. ϱ fŽ. zf Ž .x Ž. s 1 qÝ nnBn ns1 ϱ fŽ.⍀EnnnB Ž. ÝŽ.f Ž. zf Ž. x Ž. s _k00,mq q n ns1 ϱ fŽ.⍀EnnnB Ž. Ýf Ž. zf Ž. 5x Ž. s _k00,mq q n ns1 fŽ. if ⍀ E g _ k00,m F ½fŽ.Žn 1q␦1 .if g Bn fŽ. if ⍀ E 1 g _ k00,m s ½f11Ž.Ž1q␦ . if g Bn. Ž. Ž. Ž.Ž. Therefore 5f11 q zg 5F 1 q ␦ 5f1 5. Thus 55f111qzg ppF Ž.1 q ␦ 55f ␦ FŽ.1q␦11q ž/3 ␦ -1q for all z with < 555555fqzg ppF f y f11q f q zg pŽ.since p G 1 ␦␦ Fq1q for all z with < Ž.p Ž.p Case 2. 0 - p - 1. Choose ␦ e ) 0 such that ␦00r3 q 1 q ␦ r2 F Ž.p Ž. 1q␦. Argue as in Case 1 to get a function f01s f g Lp, X such pŽ.p p p p that 55f00y f ppF ␦ r3.Then 55f 0F55fpq55f 0yfpF1q Ž.p Ž.pŽŽ .p.Ž.p ␦02r3 . 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