Constructions of Uniformly Convex Functions

Transcription

1 Costructios o Uiormly Covex Fuctios Joatha M. Borwei ad Jo Vaderwer Abstract. We give precise coditios uder which the compositio o a orm with a covex uctio yields a uiormly covex uctio o a Baach space. Various applicatios are give to uctios o power type. The results are dualized to study uiorm smoothess ad several examples are provided. Key words: Covex Fuctio, uiormly covex uctio, uiormly smooth uctio, power type, Fechel cojugate, compositio, orm. 000 Mathematics Subject Classiicatio: Primary 5A4; Secodary 46G05, 46N0, 49J50, 90C5. Itroductio ad prelimiary results We work i a real Baach space X whose closed uit ball is deoted by B X, ad whose uit sphere is deoted by S X. By a proper uctio : X (, + ], we mea a uctio which is somewhere real-valued. A proper uctio : X (, + ] is covex i (λx + ( λy λ(x + ( λy or all x, y dom, 0 λ. The cojugate uctio o : X (, + ] is deied or x X by (x = sup x, x (x. x X Relevat backgroud material o covex aalysis ca be oud i various ie texts such as [9, ] ad i our ow book [3]. Give a proper covex uctio : X (, + ], its modulus o covexity is the uctio δ : [0, + [0, + ] deied by δ (t := i { (x + (y ( x + y } : x y = t, x, y dom, where the iimum over the empty set is +. We say that is uiormly covex whe δ (t > 0 or all t > 0, ad has a modulus o covexity o power type p i there exists C > 0 so that δ (t Ct p or all t > 0. I [], uiormly covex uctios are deied usig the gage o uiorm covexity, ad it ollows rom [0, Remark.] that the deiitio preseted here is equivalet to that used i [0, ]. Departmet o Mathematical Scieces, Newcastle Uiversity, NSW, Australia jborwei@ewcastle.edu.au Research supported by the Australia Research Coucil Departmet o Mathematics, La Sierra Uiversity, Riverside, CA. jvaderw@lasierra.edu.

2 Relatedly, the modulus o covexity o the orm, δ, is deied or 0 ɛ by { } δ (ɛ := i x + y : x = y =, ad x y ɛ. I the case δ (ɛ > 0 or each ɛ > 0, is said to be uiormly covex (as a orm. I there exists C > 0 ad p such that δ (ɛ Cɛ p or all 0 ɛ, the is said to have modulus o covexity o power type p. A systematic expositio o uiormly covex orms ca be oud i [6, Sectios IV.4, IV.5], ad [, Sectio 3.5] presets a thorough accout o uiormly covex uctios. However, explicit costructios o such uctios, especially those derived rom a uiormly covex orm, appear to be somewhat sparse. While it is easy to see, or example, that := r with r > is uiormly covex o bouded sets whe is uiormly covex, it is ot ecessarily globally uiormly covex. I act [] shows whe r, that is uiormly covex i ad oly i has modulus o covexity o power type r. Our goal i this ote is provide precise coditios uder which g is uiormly covex whe g is a odecreasig covex uctio o [0,. I may algorithms, uiorm covexity o bouded sets ad other weaker orms o covexity suice or their implemetatio as ca be see, or example, i [4, 5]. Noetheless, beyod their theoretical iterest, uiormly covex uctios are dual uder cojugatio to uiormly smooth covex uctios []. Also whe cosidered with moduli o power type, there is a tight duality with Hölder cotiuity coditios o the derivatives (see [, Theorem 3.5.0, Corollary 3.5. ad Theorem 3.5.]. Because uiormly covex orms, ad eve those with some power type are (abudatly available o superrelexive spaces as is discussed i the moograph [6], we believe it is importat to id explicit coditios uder which the compositio with a orm yields a uiormly covex uctio (or eve better, oe with modulus o power type. Iter alia, we adumbrate the somewhat subtle relatioship betwee otios o uiorm covexity or orms based o behaviour o the sphere ad those or covex uctios. We will use the ollowig simple examples o uiormly covex uctios o the real lie recorded i [3, Exercise 5.4.]. Fact.. Suppose a uctio o R satisies ( α > 0 o [a, where is a ixed iteger, ad that (k 0 o [a, or k {,..., +}. Deie the uctio g by g(x := (x or x a ad g(x := + or x < a. The g is uiormly covex with modulus o covexity o power type. I particular, let b > ad g(x := b x or x 0, ad g(x := + otherwise. The g is uiormly covex with modulus o covexity o power type p or ay p. Similarly, usig Taylor series oe ca show that or p ad g(x := x p or x 0, ad g(x := + otherwise, g is uiormly covex with modulus o covexity o power type p. Costructios o Uiormly Covex Fuctios Our irst objective is to determie precisely whe a compositio with a orm yields a uiormly covex uctio. Theorem.. Suppose : [0, + [0, + is covex ad odecreasig. The is uiormly covex i ad oly i ad are uiormly covex while (. lim i +(t δ t > 0 t t or each ɛ > 0.

3 Proo. : Clearly is uiormly covex because or ixed x 0 S X, we have that (t = ( tx 0 ad so is a uiormly covex uctio. Similarly, is uiormly covex. Ideed, suppose x = y = ad x + y. The ( x + ( ( y x + y 0 because is cotiuous at. The uiorm covexity o implies x y 0; thus is uiormly covex. ( Thece, suppose or some ɛ > 0 ad t that lim +(t δ ɛ t t = 0. Now choose u, v S X such that u v ɛ but u + v t δ Let x := t u ad y := t v. The x y ɛ or all, but ( ( t u + t v ɛ ( t u t δ t +(t ( ɛ ( t u ɛ where ɛ = t δ t +(t 0, which cotradicts the uiorm covexity o. : Suppose or each ɛ > 0, lim i +(t δ t > 0, ad ad are uiormly t t covex. Suppose is ot uiormly covex. The there exist (x, (y X ad ɛ > 0 such that x y ɛ or all N, but (. ( x + ( ( y x + y 0. We shall cosider various cases. First suppose lim sup x y > 0. By switchig roles o x ad y as ecessary, ad passig to a subsequece we may assume x y η > 0 or all N. Thus usig the act is odecreasig ad uiormly covex we have ( x + ( y ( x + y t. ( x + ( ( y x + y δ (η > 0 or all N. This cotradicts (.. Thus, or the rest o the proo we may ad do suppose ( x y 0. (a Cosider the case where (x is a bouded sequece. By passig to a subsequece as ecessary we may assume x α ad y α or some α 0. Because x y ɛ it is clear that α > 0 ad by the uiorm covexity o we obtai lim sup x + y α δ. α Cosequetly, lim sup x + y ] [ α δ. Usig the act that is covex ad icreasig, we obtai α lim i ( x + ( ( ( ( y x + y lim i x + y x + y ( (α α αδ > 0. α 3

4 which cotradicts (.. (b It remais to cosider the case where (x is ubouded. I act, ay bouded subsequece o (x would yield a cotradictio as above, so we let α := x ad assume α. Further, because we ow kow that ( x y 0, iterchagig x ad y as ecessary, we write y = β where α = β + η ad η 0 +. Now let x := α x ad ỹ := β y. The x ỹ ɛ η α. Fix N N such that x ỹ or N. The uiorm covexity o esures that ɛ β ( x + ỹ ɛ δ β or N. Let (.3 β := β + α ( ɛ δ β. β Note that x + y β x + ỹ + η, ad that β /β (sice β, η 0. The, or N, mootoicity o esures that ( ( x + y x + ỹ β + η ( ( ɛ β δ β + η β (.4 The covexity o guaratees that (.5 (α + ( (β α + β = ( β. ( ( β ɛ + δ β β +( β, or N. Hece (.6 ( β (α + ( ɛ (β δ β = ( x + ( y δ β β +( β β +( β. To complete the proo, it remais to veriy that ( ɛ (.7 lim i δ β β +( β > 0 ad as a cosequece it will ollow that (.6 cotradicts (.. Ideed, sice β /β, or suicietly large, β β ad because δ is odecreasig o [0, ] this additioally esures ( δ ɛ β δ or such. Cosequetly, 4 β ( ɛ δ β β +( β δ 4 β β +( β or suicietly large. Applyig (. with ɛ/4 replacig ɛ to the right-had side o the previous iequality, oe deduces (.7 as desired. 4

5 It is perhaps surprisig that the previous result meas that the compositio o a uiormly covex orm with a odecreasig uiormly covex uctio o the positive axis is a uiormly covex uctio i ad oly i (. holds. Theorem. also eables us to costruct cotiuous uiormly covex uctios usig ay uiormly covex orm o a superrelexive Baach space. Example.. Let be a uiormly covex orm with modulus δ. We deie (t := t or 0 t while t (t := t + δ (u du whe t >. We may apply Theorem. to show is uiormly covex. Proo. Ideed, is oegative icreasig o [0, so is covex ad odecreasig. Moreover, t t is uiormly covex (hece so is its sum with aother covex uctio ad so is uiormly covex. Fially, or t >, (t = t + /δ (t. For ixed ɛ whe t < ɛ we the have +(t δ t > t δ(t δ t > t t ad so (. holds. Further examples will be give ater the ollowig more qualitative result cocerig moduli o power type. Theorem.3. Suppose : [0, + [0, + is a covex odecreasig uctio ad p. (a Suppose ad have moduli o covexity o power type p ad +(t Ct p or some C > 0 ad or all t > 0. The also has modulus o covexity o power type p. (b Coversely, i has modulus o covexity o power type p, the ad have moduli o covexity o power type p. I the case that additioally satisies (.8 0 < lim i ɛ 0 + δ (ɛ ɛ p < (i.e., the modulus o is o better tha power type p, the or some K > 0 ad or all t > 0. +(t Kt p Proo. (a First we ix positive costats A, B correspodig to the respective moduli, ad let C > 0 be as give. That is, δ (ɛ Aɛ p or all ɛ > 0, δ (ɛ Bɛ p or all 0 ɛ, ad +(t Ct p or all t > 0. Let ɛ > 0 be ixed, ad suppose x, y X satisy x y ɛ. We may assume y x. Suppose irst, y + ɛ/ x. Usig the modulus o covexity o we obtai (.9 ( x + ( ( y x + y ( x + ( x + y p ( y A. Thus, or the remaider o the proo we will assume y + ɛ/ > x. 5

6 Let a := y ad x := x/ x, ỹ := y/ y. The y a x > ɛ/. Cosequetly, ỹ x > ɛ a. Thece the modulus o covexity implies x + ỹ p B a ad so (.0 ( x + y a x + ỹ + x a x + p y Ba. a (i We cosider the case, Ba p a a/. Recallig that x + y x y ɛ, we have y ɛ/4 sice y x ɛ/. Because a = y, it ollows that a/ ɛ/8. Thus, lettig t 0 := ( x + y / a/, we have t 0 a/ ad the odecreasig property o esures ( x + y (t 0. Now we use this with the covexity o to compute, ( x + ( x + y ( y (t 0 + +(t 0 (a/ (. (t 0 + +(a/ (a/ (t 0 + +(ɛ/8 (ɛ/8 ( x + y ( ɛ p + C. 8 (ii For our remaiig case, we suppose Ba p a a/. The the right had side o (.0 is at least a/. Now use the act (t C(a/ p whe t a/ to compute ( x + y ( x + y p ( a p Ba C a (. ( x + ( y BC 4 p. Puttig (.9, (. ad (. together we see that has modulus o covexity o power type p as desired. (b Because has modulus o covexity o power type p, oe eed oly ix x 0 S X ad cosider (t = ( tx 0 or t 0 to see that has modulus o covexity o power type p. Also, let β := +( ad let C > 0 be such that δ (ɛ Cɛ p whe ɛ > 0. Fix ɛ (0, ], ad choose x, y S X with x y ɛ ad x+y ( x + y ( Cɛ p = Cɛ p δ (ɛ. The ( x + y ( βδ (ɛ ad it ollows δ (ɛ C β ɛp. Thus has modulus o covexity o power type p as desired. It remais to veriy (t Mt p or some M > 0 ad all t > 0 whe (.8 is valid. Ideed, i this case, we id (u, (v S X ad K > 0 such that ɛ := u v 0 + ad u + v Kɛp. 6

7 Now ix t > 0, ad let x := tu ad y := tv. The (.3 x + y t( Kɛp. The x y = tɛ ad the modulus o covexity o implies ( x + y (.4 ( x + ( y C(tɛ p = (t Ct p ɛ p. The covexity o implies that (t tkɛ p (t +(t(tkɛ p. Usig this alog with (.3 ad the act is odecreasig, we obtai ( x + y (.5 (t tkɛ p (t +(t(tkɛ p. Combiig (.4 ad (.5 implies +(t C K tp, as desired. The ollowig corollary recovers a result rom [] whose origial proo proceeded via establishig uiorm smoothess ad ivokig duality results rom []. Corollary.4 (Theorem.3, []. Let := p where p. The the ollowig are equivalet: (a is uiormly covex; (b has modulus o covexity o power type p; (c has modulus o covexity o power type p. Proo. (a (b: Suppose is uiormly covex, the (. holds with ɛ =. Cosequetly, lim i t ptp δ (t > 0 ad so there exists C > 0 so that p t p δ (t > C or t > t 0. I particular, or 0 < ɛ < /t 0, we have δ (ɛ > Kɛ p where K := Cp. (b (c: Follows rom Theorem.3 because the uctio t t p has modulus o covexity o power type p. (c (a: is trivial. Example.5. Suppose that b > ad has modulus o covexity o power type p where p. The := b is uiormly covex with modulus o covexity o power type p. However, eve o R there are uiormly covex orms so that h := b is ot uiormly covex. Proo. Let g(t := b t. The g (t Ct p or some C > 0 ad all t 0, ad g has modulus o covexity o power type p. Accordig to Theorem.3, has modulus o covexity o power type p. For the claim cocerig h, we appeal to [7] to obtai a orm o R so that lim i t t bt log(bδ (t = 0. The (. ails, ad so Theorem. esures h is ot uiormly covex. 7

8 Oe may view the above coditios dually. For this, let us recall the modulus o smoothess o a orm is deied or τ > 0 by { } x + τh + x τh ρ (τ := sup : x = h =. Give < q, we will say has modulus o smoothess o power type q i there exists C > 0 so that ρ (τ Cτ q or τ > 0; see [6] or urther iormatio. Similarly, the modulus o smoothess o a covex uctio is deied or τ 0 by { ρ (τ := sup (x + τh + } (x τh (x : x X, h = ; as with orms, whe ρ (τ Cτ q or some C > 0 ad all τ > 0 we will say has modulus o smoothess o power type q. See [, p. 04] or [3, Sectio 5.4] or urther discussio o this ad related cocepts. We ote also that give h :=, the the cojugate is give by h (φ = sup φ(x ( x = sup φ x ( x = ( φ. x X x X We may ow preset the ollowig dual versio o Theorem.3. Corollary.6. Suppose is uiormly smooth with modulus o smoothess o power type q where < q, is odecreasig ad has modulus o smoothess o power type q while +(t Ct q, t 0. The has modulus o smoothess o power type q. Coversely, suppose has modulus o smoothess o power type q. The has modulus o smoothess o power type q, has modulus o smoothess o power type q, ad i the modulus o smoothess o is ot better tha power type q, the (t Ct q. Proo. We may assume (0 = 0 ad (0 = 0 (by subtractig the derivative at 0. Thus, we may urther assume (t = 0 or t 0. Cosequetly is odecreasig, ad (0 = 0. Let h := as above. Accordig to [6, Propositio IV..], the dual orm has modulus o covexity o power type p. Now let t (y. The t 0, y (t ad so y Ct q. Thus t Ky /(q, or equivaletly t Ky p. This implies +(y Ky p or all y 0. t 0, ad has modulus o covexity o power type p. Accordig to Theorem.3(a h has modulus o covexity o power type p. By duality, see [, Corollary 3.5.], h has modulus o smoothess o power type q. The details o the coverse ollow similarly rom Theorem.3(b; agai by ivokig duality results o [, Corollary 3.5.] ad [6, Propositio IV..]. I coclusio, we should also metio that [] provides reormigs with moduli o covexity o power type based o growth rates o uiormly covex uctios o the space. I act, [, Theorem 3.7] ca be used as ollows to illustrate the restrictiveess o obtaiig uctios that are simultaeously uiormly covex ad uiormly smooth. Remark.7. Suppose X is a Baach space ad : X R is both uiormly covex ad uiormly smooth. The X is isomorphic to a Hilbert space. Moreover, g := p is simultaeously uiormly covex ad uiormly smooth i ad oly i p = ad has modulus o smoothess ad modulus o covexity both o power type. 8

9 Proo. Let be as give. The [, Propositio 3.5.8] implies that (x lim i x x > 0. Because cotiuous covex uctios are bouded below o bouded sets, we have 4a +b or some a > 0 ad b R. Thus by replacig with b, we may assume 4a. The a ad so is uiormly covex [, Theorem 3.5.]. Accordig to [, Theorem 3.7], X admits a orm with modulus o covexity o power type. Arguig similarly with, oe ca show that B A or some A > 0 ad costat B. Applyig [, Theorem 3.7] shows that X admits a orm with modulus o covexity o power type. It ollows rom [6, Propositios IV.., IV.5.0, IV.5.] that X has type ad cotype ad so X is isomorphic to a Hilbert space by Kwapie s theorem [8]. For the moreover assertio, we ote that the oly i claim ollows rom Corollaries.4 ad.6. For the i assertio, as i the previous paragraph, the duality results just cited imply that ad are both uiormly covex ad hece [, Propositio 3.5.8] implies both that p ad that its cojugate idex q ; cosequetly, p = as claimed. Reereces [] D. Azé ad J.-P. Peot. Uiormly covex ad uiormly smooth covex uctios. A. Fac. Sci. Toulouse Math., 4: , 995. [] J. M. Borwei, A. Guirao, P. Hájek, ad J. Vaderwer. Uiormly covex uctios o Baach spaces. Proc. Amer. Math. Soc., 37:08 09, 009. [3] J. M. Borwei ad J. Vaderwer. Covex Fuctios: Costructios, Characterizatios ad Couterexamples, volume 09 o Ecyclopedia o Mathematics ad Applicatios. Cambridge Uiversity Press, 009. [4] D. Butariu ad A. N. Iusem. Totally Covex Fuctios or Fixed Poits Computatio ad Iiite Dimesioal Optimizatio. Kluwer, 000. [5] D. Butariu ad E. Resmerita. Bregma distaces, totally covex uctios, ad a method or solvig operator equatios i Baach spaces. Abstract ad Applied Aalysis, Volume 006:Ar. ID 8499, 39pp, 006. [6] R. Deville, G. Goderoy, ad V. Zizler. Smoothess ad Reormigs i Baach spaces, volume 64 o Pitma Moographs ad Surveys i Pure ad Applied Mathematics. Logma Scietiic & Techical, Harlow, 993. [7] A. Guirao ad P. Hájek. O the moduli o covexity. Proc. Amer. Math. Soc., 35: , 007. [8] S. Kwapie. Isomorphic characterizatios o ier product space by orthogoal series with vector valued coeiciets. Studia Math., 44: , 97. [9] R. T. Rockaellar. Covex Aalysis. Priceto Uiversity Press, Priceto, New Jersey,

10 [0] C. Zăliescu. O uiormly covex uctios. J. Math. Aal. Appl., 95: , 983. [] C. Zăliescu. Covex Aalysis i Geeral Vector Spaces. World Scietiic, New Jersey- Lodo-Sigapore-Hog Kog, 00. 0