On continuous choice of retractions onto nonconvex subsets

ON CONTINUOUS CHOICE OF RETRA CTIONS ONTO NONCONVE X SUBSETS Du ˇ san Repo v ˇ s and P a vel V. Semenov Abstract. F or a Banac h space B and for a class A of its b ounded closed retracts, endo wed with th e Hausdorff met ric, we prov e that retractions on elements A ∈ A can b e chosen to depend con tin uously on A , wheneve r noncon vexit y of each A ∈ A is less than 1 2 . The key ge ometric argument is t hat the set of all uniform retractions on to an α − paracon vex set (in the spirit of E. Michael) is α 1 − α − paracon vex s ubset in the space of con tinuous mappings of B into itself. F or a Hilber t space H the estimate α 1 − α can b e i mprov ed to α (1+ α 2 ) 1 − α 2 and the constan t 1 2 can b e r educed to the ro ot of the equation α + α 2 + a 3 = 1. 0. In tro ductio n The initial source of the pres ent pap er was tw o-fold. Probably it was Bing [12 ] who first asked whether there exists a co nt inuous function which selects a p o int from each a r c of the E uclidean plane. Hams tr¨ om and Dyer [3] observed that this problem reduces to the pro blem o f contin uous c hoice of retr actions o nto arcs. In fact, it suffices to co ns ider the images o f a chosen p oint with resp ect to co nt inuously chosen retractions . A simple construction ba s ed, for example, on the sin( 1 x ) − curve shows that in general there ar e no contin uously chosen retractions for the family of arcs top ologized by the Ha us dorff metric. Therefore a strong er top ology is nee de d for an affir mative answer. In fact, for any homeomorphic compact subsets A 1 and A 2 of a metric s pace B o ne can consider the so-ca lled h − metric d h ( A 1 , A 2 ) defined b y : d h ( A 1 , A 2 ) = sup { dist ( x, h ( x )) : h runs over all homeomorphisms of A 1 onto A 2 } and consider the c ompletely r e gular top olog y on the family of all suba rcs, gene r ated by s uch a metric. With r esp ect to this topo logy , P ixley [12] a ffirmatively resolved the pr oblem of contin uo us choice for r e tr actions onto subarc s in an a rbitrar y sepa- rable metric space. By returning to the more s tandard Hausdorff top olog y in the subspace exp AR ( B ) of a ll compact absolute retracts in B o ne can try to sea rch for a degr ee of nonco n- vexit y of s uch a r etracts. In the s implest situatio n, for co nv ex exp onent e xp conv ( B ) 1991 Mathematics Subject Classific ation . Primar y: 54C60, 54C65, 41A65. Secondary: 54C55, 54C20. Key wor ds and phr ases. Contin uous retractions, contin uous selections, paraconv exity , Banac h spaces, low er s emicon tinuous multiv alued mappings. The first author was supp orted by the SRA gr an ts P1-0292-0101-04, J1-9643-0101 and BI- RU/0 8-09-002. The second author was supp orted by the RFBR grant 08-01-00663. W e thank the referee for comments and suggestions. Ty peset b y A M S -T E X 1 2 DU ˇ SAN REPOV ˇ S AND P A VEL V. SEMENOV consisting of a ll compact conv ex s ubsets of B , cont inuous choice o f r e tr actions is a direct corolla ry o f the follo wing Michael theorem [8]: Con vex-V alued Selection Theorem. Any mu ltivalue d ma pping F : X → Y admits a c ontinuous singlevalue d sele ct ion f : X → Y , f ( x ) ∈ F ( x ) , pr ovide d that: (1) X is a p ar ac omp act sp ac e; (2) Y is a Banach sp ac e; (3) F is a lower semic ont inuous (LSC) mappi ng; (4) F or every x ∈ X , F ( x ) is a nonempty c onvex subset of Y ; and (5) F or every x ∈ X , F ( x ) is a close d subset of Y . In fact, let X = exp conv ( B ), let Y be the space C b ( B , B ) of all contin uous bo unded ma ppings of B into itself and s uppo se that F : X → Y asso ciates to ea ch A ∈ X the nonempty set of all retra ctions o f B onto A . T hen a ll hypotheses (1) − (5) can b e verified and the conclusio n of the theor em gives the desir ed contin uo usly chosen retra ctions. How ever, wha t ca n one say ab out no nconv ex absolute retracts? In g e neral, there exists an entire branch o f ma thematics devoted to v arious g eneraliza tio ns and ver- sions o f the co nv exity . In our opinion, even if one simply lists the titles of ” gener- alized conv e x ities” o ne will find a s a minimum, nearly 20 different notions. Among them ar e Me ng er’s metric conv e xity [7], Le v y’s abstract conv exity [5], Michael’s conv ex structures [9 ], Pro dano v’s alg ebraic conv exity [13], M¨ ag e rl’s pav ed spaces [6], v an de V el’s top olog ic al conv ex ity [21], decomp osable sets [1], Belya ws k i’s sim- plicial conv exity [2], Horv ath’s structures [4], Saveliev’s co nv exity [18 ], and ma ny others. Typically , a creation of ”genera liz ed conv exities”, is us ually rela ted to an extra c- tion of several principa l prop erties of the classical c o nv exity which ar e used in one of the key mathematical theor ems or theorie s and, co nsequently deals with analysis and gener alization of these prop erties in maxima lly p ossible general settings. Ba sed on the ingenious idea of Mic hael who prop osed in [10] the notion of a paraconv ex set, the authors of [14-17 , 19 ] sy stematically studied another approa ch to weak e n- ing or controlled omission of conv ex ity on a set of principal theorems of m ultiv alued analysis and top olog y . Roughly spea k ing, to each closed subset P ⊂ B of a Ba- nach space we have ass o ciated a numerical function, say α P : (0 , + ∞ ) → [0 , 2), the so-called function of nonconvexit y of P . The iden tity α P ≡ 0 is equiv ale nt to the conv exity of P and the more α P differs from zer o the ” less co nv ex” is the set P . Such cla ssical r esults ab out m ultiv a lued mappings as the Michael selection theo - rem, the Cellina approximation theorem, the Kakutani-Glicksbe r g fixed p oint theo- rem, the von Neumann - Sion minimax theor em, e tc. ar e v alid with the re placement of the conv exity assumption for v alues F ( x ) , x ∈ X of a mapping F by some a p- propriate control o f their functions of nonconv e x it y . In comparison with usual ideas of ”gener alized co nv exities”, we never define in this appr oach, for example, a ” generalized seg ment ” joining x ∈ P and y ∈ P . W e lo ok only for the distances b etw een points z o f the classic al segment [ x, y ] and the set P and lo o k for the ra tio of these distances and the s ize of the s egment. So the following na tural question immediately arises: Do es para conv exity o f a se t with r esp ect to the classical co nv exity structur e coincide with conv exity under some generalized conv exity s tructure? Corolla ries 2.5 a nd 2.6, ba sed on co ntin uous choice of a retra ction, in pa rticular provide an affir mative a nswer. ON CONTINUOUS CHOICE OF RETRA CTIONS 3 1. Preliminaries Below we denote by D ( c, r ) the o p en ball centered at the p oint c with the radius r and de no te by D r an arbitrar y op en ball with the radius r in a metr ic space. So for a nonempty subset P ⊂ Y of a no rmed space Y , a nd for a n op en r -ball D r ⊂ Y w e define the relative prec is ion of an approximation of P by elemen ts of D r as follows: δ ( P , D r ) = sup  dist ( q , P ) r : q ∈ conv ( P ∩ D r )  . F or a nonempty subset P ⊂ Y of a normed space Y the function α P ( · ) of nonc onvexity of P a sso ciates to each p os itive num b er r the followin g nonnegative nu mber α P ( r ) = sup { δ ( P, D r ) | D r is an op en r -ba ll } . Clearly , the identit y α P ( · ) ≡ 0 is equiv a lent to the c onvexity of the clo sed set P . Definition 1 .1. F or a nonn e gative nu mb er α t he nonempty close d s et P is said to b e α -p ar ac onvex, whenever α major ates the function α P ( · ) of nonc onvexity of t he set, i.e dist ( q , P ) ≤ α · r, ∀ q ∈ conv ( P ∩ D r ) . The nonempty close d set P is said to b e p ar ac onvex if it is α - p ar ac onvex for some α < 1 . Recall, that a multiv alued mapping F : X → Y b etw een top olog ical spaces is called lower semic ontinu ous (LSC for shor tness) if for each op en U ⊂ Y , its full preimage, i.e. the set F − 1 ( U ) = { x ∈ X | F ( x ) ∩ U 6 = ∅ } is op en in X . Recall also that a sing lev alued mapping f : X → Y is called a sele ction (resp. an ε - sele ction ) of a multiv alued mapping F : X → Y if f ( x ) ∈ F ( x ) (re sp. dist ( f ( x ) , F ( x )) < ε ), for all x ∈ X . Mich ael [9] pr ov ed the following selection theorem: P aracon vex-V alued Selection Theorem. F or e ach numb er 0 ≤ α < 1 any mul- tivalue d mapping F : X → Y admits a c ontinuous singlevalue d sele ct ion whenever: (1) X is a p ar ac omp act sp ac e; (2) Y is a Banach sp ac e; (3) F is a lower semic ont inuous (LSC) mappi ng; and (4) al l values F ( x ) , x ∈ X ar e α − p ar ac onvex. As a corollar y , every α − paraco nv ex set, α < 1, is con tractible and moreover, it is an absolute extensor ( AE ) with resp ect to the class of a ll para compact spaces. Hence, it is an absolute retra ct ( AR ). Moreover by [1 7], metric ε -neighborho o d of a para conv ex se t in any unifor mly conv ex Bana ch space Y , is also a parac onv ex set, and hence is a lso a n AR . F or each n um be r 0 ≤ α < 1 we denote by exp α ( B ) the family of all α − parac o nv ex compact s ubsets a nd by be xp α ( B ) the family of all α − par aconv ex b ounded subsets of a Banach space B endowed with the Hausdo r ff metr ic. Recall that the Hausdo rff 4 DU ˇ SAN REPOV ˇ S AND P A VEL V. SEMENOV distance b etw ee n tw o b o unded sets is defined as the infimum of the set of all ε > 0 such that each of the se ts is a subset of an op en ε − neighborho od of the other s et. F or ea ch retra ct A ⊂ B w e denote by Re tr ( A ) the set of all contin uous r e tractions of B o nt o A . So the multiv a lue d mapping R e tr as so ciates to each retract A ⊂ B the set of all retrac tio ns of B o nt o A . F o r chec king of the low e r semicontin uity of a mappings into the spaces of re tractions and for proving paraconv exity of these spaces we also need the notio n of an uniform retra ction (in termino lo gy of [1 1]), or a r e gular re traction (in terminology of [20])). Recall that a cont inuous re tr action R : B → A is said to be uniform (with r e s pe ct to A ) if ∀ ε > 0 ∃ δ > 0 ∀ x ∈ B : dist ( x, A ) < δ ⇒ dist ( x, R ( x )) < ε. W e emphasize that a uniform r etraction in g e ne r al is not a uniform mapping in the classical metric s e ns e. Clearly , each contin uous r etraction onto a c omp act subset is uniform with res pe ct to the set. So we denote by U Retr ( A ) the set of all co ntin uous retractions of B on to A which ar e uniform with resp ect to A . 2. The Banac h space case Theorem 2.0. L et 0 ≤ α < 1 2 and F : X → bexp α ( B ) b e a c ont inuous multivalue d mapping of a p ar ac omp act sp ac e X int o a Banach sp ac e B . Then ther e exists a c ontinuous singlevalue d mapping F : X → C b ( B , B ) such that for every x ∈ X the mapping F x : B → B is a c ontinuous r etr action of B onto the value F ( x ) of F . Sketch of pr o of of The or em 2.0. P rop osition 2.4 b elow is a coro llary of the Para- conv ex- v a lued s election theor em due to Prop os itions 2.1-2.3 and the fact that 0 ≤ α 1 − α < 1 ⇔ 0 ≤ α < 1 2 . In turn, Theorem 2.0. follows directly from Prop os ition 2.4, it suffices to put F = R ◦ F . Prop ositi o n 2.1. F or every 0 ≤ α < 1 and for e ach b ounde d α − p ar ac onvex su bset P t he set U R e tr ( P ) is a nonempty close d su bset of C b ( B , B ) . Prop ositi o n 2.2. F or every 0 ≤ α < 1 and and for every b oun de d α − p ar ac onvex subset P ⊂ B t he s et U Retr ( P ) is an α 1 − α − p ar ac onvex subset of C b ( B , B ) . Prop ositi o n 2.3 . F or every 0 ≤ α < 1 the r estriction U R etr | bexp α ( B ) , P 7→ U Retr ( P ) is lower s emic ontinuous. Prop ositi o n 2.4. F or every 0 ≤ α < 1 2 the r est riction U Re tr | bexp α ( B ) , P 7→ U Retr ( P ) admits a singlevalue d c ont inuous sele ct ion, say R : bexp α ( B ) → C b ( B , B ) , R P ∈ U Retr ( P ) . Pr o of of Pr op osition 2.1. Cle a rly for each bo unded closed retra ct A the sets Retr ( A ) and U Re tr ( A ) are clo sed in the Bana ch space C b ( B , B ). T o o btain the nonempti- ness of Retr ( P ) for the α − pa raconv ex set P it suffices to apply the Paraconv ex - v alued s election theorem to the mapping F : B → B defined by setting F ( x ) = P for x ∈ B \ P and F ( x ) = { x } for x ∈ P . T o c o nstruct a uniform re traction R : B → P one must study more in detail the idea o f the pro of of the Paraco nv ex- v alued selection theor em. Let us denote by d ( x ) the distance betw ee n a p oint x ∈ B and a fixed α − par a- conv ex subs e t P ⊂ B . F or every x ∈ B \ P fir st consider the intersection o f the set ON CONTINUOUS CHOICE OF RETRA CTIONS 5 P with the op en ball D ( x, 2 d ( x )). Next, take the convex hull conv { P ∩ D ( x, 2 d ( x )) } and finally , define the con vex-v alued ma pping H 1 : B \ P → B b y s etting H 1 ( x ) = conv { P ∩ D ( x, 2 d ( x )) } . This mapping is a LSC mapping defined on the paraco mpact domain B \ P with nonempty closed conv ex v alues in Ba nach space. So the Conv ex-v a lued selection theorem g uarantees the existence of a co ntin uous singlev alued selec tion, say h 1 : B \ P → B , h 1 ( x ) ∈ H 1 ( x ). F or an arbitra ry α < β < 1 the α − pa raconv exity of P implies the inequalities dist ( h 1 ( x ) , P ) < β · 2 d ( x ) , dist ( x, h 1 ( x )) ≤ 2 d ( x ) , x ∈ B \ P . Similarly , define the conv ex-v alued and clo s ed-v alued LSC mapping H 2 : B \ P → B by s e tting H 2 ( x ) = conv { P ∩ D ( h 1 ( x ) , β · 2 d ( x )) } , x ∈ B \ P . F or its co n tinu ous singlev alued selection h 2 : B \ P → B , h 2 ( x ) ∈ H 2 ( x ) we s ee that for every x ∈ B \ P , dist ( h 2 ( x ) , P ) ≤ α · β · 2 d ( x ) < β 2 · 2 d ( x ) , dist ( h 2 ( x ) , h 1 ( x )) ≤ β · 2 d ( x ) , once again due to the α − paraconvexit y of P . One can inductively constr uct a sequence { h n } ∞ n =1 of contin uous sing lev alued mappings h n : B \ P → B suc h that for every x ∈ B \ P , dist ( h n +1 ( x ) , P ) < β n +1 · 2 d ( x ) , dist ( h n +1 ( x ) , h n ( x )) ≤ β n · 2 d ( x ) . So the seq uenc e { h n } ∞ n =1 is lo ca lly unifor mly convergen t a nd hence its p oint wise limit h ( x ) = li m n →∞ h n ( x ) is well-defined and contin uous. Moreov er, h ( x ) ∈ P, x ∈ B \ P , due to the clos edness of P and conv erg ency of { h n } ∞ n =1 . Hence the mapping R : B → P defined by R ( x ) = h ( x ) , x ∈ B \ P a nd R ( x ) = x, x ∈ P is a retra ction of B onto P w hich is contin uo us ov er the set B \ P by construction. T o finish the pro o f we es timate tha t for every x ∈ B \ P : dist ( x, h ( x )) ≤ dist ( x, h 1 ( x )) + ∞ X n =1 dist ( h n ( x ) , h n +1 ( x )) ≤ ≤ 2 d ( x )(1 + β + β 2 + β 3 + ... ) = C · d ( x ) , for the constant C = 2 1 − β . So for x 0 ∈ P and fo r x ∈ B \ P we hav e dist ( R ( x 0 ) , R ( x )) = dist ( x 0 , h ( x )) ≤ dist ( x 0 , x ) + dist ( x, h ( x )) ≤ ≤ dist ( x 0 , x ) + C · d ( x ) ≤ (1 + C ) dist ( x 0 , x ) . The contin uity of the r etraction R : B → P over the closed subset P ⊂ B a nd its uniformity clear ly follow fr om the las t inequa lity .  6 DU ˇ SAN REPOV ˇ S AND P A VEL V. SEMENOV Pr o of of Pr op osition 2.2. Pick an o p e n ball D ( h, r ) with the ra dius r in the space C b ( B , B ) centered a t the mapping h ∈ C b ( B , B ) which intersects with the closed s et U Retr ( P ). Let R 1 , R 2 , ..., R n be elements of the in ter section D ( h, r ) ∩ U R etr ( P ) and let Q b e a co nv ex co mbination of R 1 , R 2 , ..., R n . W e wan t to estimate the distance b etw een Q and U Retr ( P ). Pick a po int x ∈ B . Passing fr om the mappings h , Q , R 1 , R 2 , ..., R n ∈ C b ( B , B ) to their v alues at x we find the op en ball D ( h ( x ) , r ) with the r adius r in the space B centered at h ( x ) ∈ B , the finite set { R 1 ( x ) , R 2 ( x ) , ..., R n ( x ) } of elements from the intersection D ( h ( x ) , r ) T P and the p oint Q ( x ) ∈ conv ( D ( h ( x ) , r ) ∩ P ). Having all fixed contin uous mappings h, Q, R 1 , R 2 , ..., R n ∈ C b ( B , B ) we see that all p oints h ( x ) , Q ( x ) , R 1 ( x ) , R 2 ( x ) , ..., R n ( x ) ∈ B con tinu ously dep end on x ∈ B . Let r ( x ) be the Chebyshev ra dius of the compact conv ex finite-dimensional set ∆( x ) = conv { R 1 ( x ) , ..., R n ( x ) } , i.e. the infim um (in fact, the minimum), of the set o f ra dii o f all clo sed ba lls containing ∆( x ). Clearly r ( x ) < r , x ∈ X . Mor eov er r ( x ) contin uously dep ends on x and for any p o sitive γ > 0 the entire set ∆( x ) lies in the op en ba ll D ( C ( x ) , r ( x ) + γ ) fo r some s uitable po int C ( x ) ∈ ∆( x ). Henceforth, the α − paraconvexit y o f P implies tha t for an arbitrary α < β the inequality dist ( Q ( x ) , P ) < β ·  ( x ) ,  ( x ) = r ( x ) + γ holds. So, the multiv alued mapping F 1 ( x ) = conv { P ∩ D ( Q ( x ) , β ·  ( x )) } . is a LSC mapping with nonempty conv ex and clos ed v alues . Note that for each x ∈ P all p oints R 1 ( x ) , R 2 ( x ) , ..., R n ( x ) , Q ( x ) co incide with x b ecause all R 1 , ..., R n are retractions o nto P . So the identit y ma pping id | P is a contin uous se le ction of F 1 | P . Therefore the mapping G 1 which is identit y on P ⊂ B and otherw is e coincides with F 1 admits a co ntin uous singlev alued selectio n, say Q 1 : B → B , Q 1 ( x ) ∈ G 1 ( x ).The α − paraco nv exity o f P and the co nstruction imply that dist ( Q 1 ( x ) , P ) < β 2 ·  ( x ) , dist ( Q 1 ( x ) , Q ( x )) ≤ β ·  ( x ) , Q 1 | P = id | P . Similarly , the multiv alued ma pping defined by setting F 2 ( x ) = conv { P ∩ D ( Q 1 ( x ) , β 2 ·  ( x )) } admits a contin uo us single v a lued s election, say Q 2 : B → B s uch that dist ( Q 2 ( x ) , P ) < β 3 ·  ( x ) , dist ( Q 2 ( x ) , Q 1 ( x )) ≤ β 2 ·  ( x ) , Q 2 | P = id | P . Inductively we obtain a sequence { Q n } ∞ n =1 of c ontin uous singlev a lued mappings Q n : B → B with the prop erties that Q n | P = id | P and dist ( Q n +1 ( x ) , P ) < β n +2 ·  ( x ) , dist ( Q n +1 ( x ) , Q n ( x )) ≤ β n +1 ·  ( x ) . ON CONTINUOUS CHOICE OF RETRA CTIONS 7 Clearly the p oint wise limit R of the sequence { Q n } ∞ n =1 is contin uo us retra ction of B o nto P a nd, mo reov e r , dist ( Q ( x ) , R ( x )) ≤ dist ( Q ( x ) , Q 1 ( x )) + ∞ X n =1 dist ( Q n ( x ) , Q n +1 ( x )) ≤ ≤ β · (1 + β + β 2 + β 3 + ... ) ·  ( x ) = β 1 − β ·  ( x ) . Hence, dist ( Q, Retr ( P )) ≤ β 1 − β ·  ( x ) = β 1 − β · ( r ( x ) + γ ) < β 1 − β · ( r + γ ) . Passing to β → α + 0 and to γ → 0 + 0 w e conclude dist ( Q, Retr ( P )) ≤ α 1 − α · r . T o finish the pr o of we must chec k that the retractio ns R ( x ) = lim n →∞ Q n ( x ) , x ∈ X onto P co nstructed ab ov e ar e uniform with r e s pe ct to P . T o this end, us ing uniformity of all retrac tio ns R 1 , ..., R n , for an arbitra ry ε > 0 c ho ose δ > 0 such that dist ( x, P ) < δ ⇒ dist ( x, R i ( x )) < ε. In par ticula r, fo r every po int x with dist ( x, P ) < δ a ll v alues R 1 ( x ) , ..., R n ( x ) , Q ( x ) are in the op en ba ll D ( x, ε ). Hence r ( x ) < ε and this is wh y dist ( x, R ( x )) ≤ dist ( x, Q ( x )) + dist ( Q ( x ) , R ( x )) < ε + β 1 − β ·  ( x ) < 1 1 − β · ( ε + γ ) Therefore R ∈ U R etr ( P ) and dist ( Q, U Retr ( P )) ≤ α 1 − α · r . So U Retr ( P ) is α 1 − α − paraco nv ex.  Pr o of of Pr op osition 2.3. Pic k P ∈ bexp α ( B ), an uniform retra c tion R ∈ U Re tr ( P ) and a num b er ε > 0 . So let δ > 0 b e such that δ < (1 − α ) · ε a nd dist ( x, P ) < δ ⇒ dist ( x, R ( x )) < (1 − α ) · ε. Consider an y P ′ ∈ be xp α ( B ) which is δ − clos e to P with re sp ect to the Haus- dorff dista nce . W e must find a uniform r etraction R ′ ∈ U Retr ( P ′ ) such that dist ( R, R ′ ) < ε. The multiv alued mapping F ′ : B → B s uch tha t F ′ ( x ) = { x } , x ∈ P ′ and F ′ ( x ) = P ′ otherwise is a LSC mapping with α − paraco nvex v alues. Any se le c tion of F ′ is a retraction o n to P ′ . So let us chec k that R is a lmost selection o f F ′ and hence, is almost a re traction onto P ′ . F or every x ∈ B \ P ′ we hav e dist ( R ( x ) , F ′ ( x )) = dist ( R ( x ) , P ′ ) < δ < (1 − α ) ε bec ause R ( x ) ∈ P and the s et P lies in the δ − neig hbo rho o d of the set P ′ . If x ∈ P ′ then dist ( R ( x ) , F ′ ( x )) = dist ( R ( x ) , x ) < (1 − α ) ε 8 DU ˇ SAN REPOV ˇ S AND P A VEL V. SEMENOV bec ause the set P ′ lies in the δ − neig hborho o d of the set P and due to the choice of the num ber δ . Hence, the retraction R of B onto the set P is a contin uo us singlev alued ε ′ − selection of the mapping F ′ , ε ′ = (1 − α ) ε . F ollowing the pro ofs of P rop ositions 2.1 and 2.2 we can improve the ε ′ − selection R of F ′ to a selection R ′ of F ′ such that di st ( R, R ′ ) < ε ′ 1 − α = ε . So R ′ is a contin- uous retr action onto P ′ and the c hecking of unifor mit y of R ′ can b e p erfo rmed by rep eating the arg ument s o n Chebyshev r adii from the pro of of P rop osition 2 .2.  Observe the pr o of of Theo rems 2.0 for the case o f compact par aconv ex sets is muc h more easier , b ecause for a n y co mpact retrac t A ⊂ B each contin uous retraction B → A automatically will b e unifor m with resp ect to A . So , one can uses directly R etr ( A ) instead of U R etr ( A ). Corollary 2.5. Under the assumptions of The or em 2 .0 if in addition al l values F ( x ) , x ∈ X , ar e p airwise disjoint t hen the m etric subsp ac e Y = S x ∈ X F ( x ) ⊂ B admits a c onvex metric structur e σ (in the sense of [9] ) such that e ach value F ( x ) is c onvex with r esp e ct to σ . Pr o of. By Theorem 2.0, let R ( x ) : B → F ( x ) , x ∈ X , be a contin uo us family of unifor m contin uous retractions o nto the v alues F ( x ). One can define a co nv ex metric structure σ on Y = S x ∈ X F ( x ) by setting that σ − c o nv ex combinations ar e defined only for finite subs e ts { y 1 , y 2 , ..., y n } which a re entirely displaced in a v alue F ( x ) and σ − conv F ( x ) { y 1 , y 2 , ..., y n } = R ( x )( conv B { y 1 , y 2 , ..., y n } ) .  Corollary 2. 6. L et f : Y → X b e a c ontinuous singlevalue d surje ction and let al l p oint- inverses f − 1 ( x ) , x ∈ X ar e α − p ar ac onvex sub c omp acta of Y with α < 1 2 . Then Y admits a c onvexity metric structure such that e ach p oint-inverse is c onvex with r esp e ct to this structur e. 3. The Hilb ert space case Hilber t spa ces hav e a many o f adv a ntages inside the class of all Bana ch spaces . In this chapter we demonstra te s uch a adv antage re la ted to paraco nvexit y . Briefly we prove the estimate α 1 − α for pa r aconv exity of the set Ret r ( P ) onto α − pa raconvex set P ca n be improv ed with α (1 + α 2 ) 1 − α 2 = α 1 − α · 1 + α 2 1 + α < α 1 − α . Hence in Theorem 2.0 one can s ubstitute the r o ot of the equation α + α 2 + α 3 = 1 instead o f 1 2 . In fact, a gener alization of s uch type can b e p erformed for any uniformly c o nv ex Ba nach spac e s but for Hilber t s pace the pr o ofs differ only in techn ical details. Theorem 3. 0. L et H b e a Hilb ert sp ac e and F : X → bexp α ( H ) b e a c ontinuous mapping of a p ar ac omp act sp ac e X , wher e α + α 2 + α 3 < 1 . Th en ther e exists a c ontinuous singlevalue d mapping F : X → C b ( H, H ) such t hat for every x ∈ X the mapping F x : H → H is a c ontinuous r etr action of H onto the value F ( x ) of F . So we rep eat the or iginal definition of α − pa raconvexit y of P but with the ap- propriate es tima te for distances b etw een po int s of simplices and p oints o f P inside op en balls. ON CONTINUOUS CHOICE OF RETRA CTIONS 9 Definition 3.1. L et 0 ≤ α < 1 . A nonempty close d subset P ⊂ B of a Banach sp ac e B is said to b e strongly α - pa raconvex if for every op en b al l D ⊂ B with r adius r and for every q ∈ conv ( P ∩ D ) the distanc e di s t ( q , P ∩ D ) is less t han or e qual to α · r . Clearly , strong α − par aconv exity o f a set implies its α − para conv exity . In a Hilber t spa ce the conv ers e is almos t true: for so me 1 > β > α , α − par aconv exity implies strong β − paraconvexit y for so me suita ble β . Prop ositi o n 3.2. Any α − p ar ac onvex subset P of a Hilb ert sp ac e is its st r ong ϕ ( α ) − p ar ac onvex subset, wher e ϕ ( α ) = p 1 − (1 − α ) 2 = √ 2 α − α 2 . Prop ositio n 3.2 is an immediate coro llary of the following purely geo metrical lemma: Lemma 3.3. L et D = D r b e an op en b al l with the r adius r in the Hilb ert sp ac e H . L et z b e a p oint of the c onvex hul l conv ( P ∩ D ) of the interse ction D with a set P and let di st ( z , P ) ≤ α · r . The n dist ( z , P ∩ D ) ≤ ϕ ( α ) · r Pr o of of L emma 3.3. P ick an arbitra ry α < γ < 1 and let c b e the cent er of the op en ball D = D ( c, r ). If dist ( c, z ) ≤ (1 − γ ) · r then the whole op en ball D ( z , γ · r ) lies inside of D . Hence, a p oint p ∈ P which is ( γ · r ) − clo se to z automatically will b e in D . So dist ( z , P ∩ D ) ≤ dist ( z , p ) < γ · r ≤ ϕ ( γ ) · r. Let us lo ok for the opp osite case when z is ” close” to the bo unda ry of the ball D , i.e. when (1 − γ ) · r < dist ( c, z ) < r . Draw the hyperplane Π supp orting to the ball D ( c, dist ( c, z )) a t the point z . Such the hyp e r plane Π divides the ball D into tw o op en convex parts: the ce n ter c b elongs to the ”lar ger” part D + whereas the po int z b elo ngs to the the b oundary o f ” s maller” pa rt D − . Clearly , C l os ( D − ) contains a po int p ∈ P (if, to the contrary , P ∩ D is subset o f D + then z ∈ conv ( P ∩ D ) ⊂ D + ). Hence, the distance dist ( z , P ∩ D ) ma jora tes by dist ( z , p ) ≤ max { dist ( z , u ) : u ∈ C l os ( D − ) } = ϕ  dist ( c, z ) r  · r < ϕ ( γ ) · r. So in b oth cases dis t ( z , P ∩ D ) ≤ ϕ ( γ ) · r a nd pa ssing to γ → α + 0 w e se e that dist ( z , P ∩ D ) ≤ ϕ ( α ) · r  Recall that for a multiv alued mapping F : X → Y and for a n umer ical function ε : X → (0 , + ∞ ) a singlev alued mapping f : X → Y is s aid to b e a n ε − selection of F if di st ( f ( x ) , F ( x )) < ε ( x ) , x ∈ X . Prop ositi o n 3. 4. L et 0 ≤ α < 1 and let F : X → H b e an α − p ar ac onvex value d LSC mapping fr om a p ar ac omp act domain into a H ilb ert sp ac e. Then (1) for e ach c onstant C > 1+ α 2 1 − α 2 , for every c ontinuous funct ion ε : X → (0 , + ∞ ) and for every c ontinuous ε − sele ct ion f ε : X → H of the mapping F ther e exists a c ontinuous sele ction f : X → H of F such that dist ( f ε ( x ) , f ( x )) < C · ε ( x ) , x ∈ X ; (2) F admits a c ont inuous sele ct ion f . 10 DU ˇ SAN REPOV ˇ S AND P A VEL V. SEMENOV Pr o of. Clea rly (1) implies (2): the mapping x 7→ [1 + dist (0 , F ( x )) , + ∞ ), x ∈ X , is a LSC mapping with nonempty clo s ed and conv ex v alue s and therefore admits a contin uous selec tio n, s ay ε : X → (0 , + ∞ ). Therefore f ε ≡ 0 is an ε -selection o f F . T o prov e (1 ) let ϕ ( t ) = √ 2 t − t 2 , 0 < t < 1, c ho ose any γ ∈ ( α, 1) and deno te by D ( x ) = D ( f ε ( x ) , ε ( x )). As ab ove, the m ultiv alued mapping F 1 ( x ) = conv { F ( x ) ∩ D ( x ) } , x ∈ X admits a singlev a lued co ntin uous s election, say f 1 : X → H . F or each x ∈ X the p oint f 1 ( x ) belo ngs to the convex hull conv { F ( x ) ∩ D ( x ) } and dist ( f 1 ( x ) , F ( x )) ≤ α · ε ( x ) due to the α − pa raconv exity of the v alue F ( x ). Lemma 3.3 implies that dist ( f 1 ( x ) , F ( x ) ∩ D ( x )) ≤ ϕ ( α ) · ε ( x ) < ϕ ( γ ) · ε ( x ) . Therefore, the multiv alued mapping F 2 : X → H defined by F 2 ( x ) = conv { F ( x ) ∩ D ( x ) ∩ D ( f 1 ( x ) , ϕ ( γ ) · ε ( x )) } , x ∈ X is a LSC mapping with nonempty closed a nd co n vex v alues. Hence there exists a selection of F 2 , say f 2 : X → H . F or each x ∈ X the p oint f 2 ( x ) belo ngs to the convex hull conv { F ( x ) ∩ D ( x ) } and dist ( f 2 ( x ) , F ( x )) ≤ α · ϕ ( γ ) · ε ( x ) due to the α − para conv exity o f the v a lue F ( x ) and bec ause f 2 ( x ) ∈ conv { F ( x ) ∩ D ( f 1 ( x ) , ϕ ( γ ) · ε ( x )) } . Lemma 3.3 implies that dist ( f 2 ( x ) , F ( x )) ≤ ϕ ( α · ϕ ( γ )) · ε ( x ) < ϕ ( γ · ϕ ( γ )) · ε ( x ) , x ∈ X . Put F 3 ( x ) = conv { F ( x ) ∩ D ( f ε ( x ) , ε ( x )) ∩ D ( f 2 ( x ) , ϕ ( γ · ϕ ( γ )) · ε ( x )) } , x ∈ X and so on. Hence w e hav e constr ucted a s equence { f n : X → H } ∞ n =1 of contin uous singlev alued mappings such that dist ( f ε ( x ) , f n ( x )) ≤ ε ( x ) , dist ( f n ( x ) , F ( x )) < γ n · ε ( x ) where γ 1 = γ and γ n +1 = γ · ϕ ( γ n ) . The sequence { γ n } is monotone, decr easing and conv er ges to the (p ositive!) ro ot of equation t = γ · ϕ ( t ), i.e. to the num b er t = 2 γ 2 1+ γ 2 > 2 α 2 1+ α 2 . Therefore w e can choose num b ers N ∈ N and λ such tha t 1 > 1 − 1 C > λ > γ N > 2 γ 2 1 + γ 2 > 2 α 2 1 + α 2 . Hence, the mapping g 1 = f N is a ( λ · ε ) − s election of F and dist ( f ε ( x )) , g 1 ( x )) ≤ ε ( x ) . Starting with g 1 one can find λ 2 · ε − selection g 2 of F such that dist ( g 1 ( x ) , g 2 ( x )) ≤ λ · ε ( x ) . Contin uation of this co nstruction pro duces a contin uous s election f = li m n →∞ g n of F such tha t dist ( f ε ( x ) , f ( x )) ≤ ε ( x ) · (1 + λ + λ 2 + ... ) = 1 1 − λ · ε ( x ) < C · ε ( x ) , x ∈ X .  Prop ositio n 3.4 implies the following analog of Pro p osition 2 .2: ON CONTINUOUS CHOICE OF RETRA CTIONS 11 Corollary 3.5. F or every 0 ≤ α < 1 and for every b ounde d α − p ar ac onvex subset P ⊂ H the set R etr ( P ) is an α (1+ α 2 ) 1 − α 2 − p ar ac onvex subset of C b ( H, H ) . Pr o of of The or em 3.0. It suffices to rep eat the pro o f of Theorem 2.0 but we use Corollar y 3.5 instead of Prop osition 2.2.  4. Concluding remarks Roughly sp eaking, we have prov ed that α − para conv exity of a set implies β − para- conv exity of a set of a ll re tractions o nt o this set with β = β ( α ) = α 1 − α . Such an es timate for β = β ( α ) naturally a pp e ars as a r esult o f the usual geometric progre s sion pro cedure. 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