On Pli{s} metric on the space of strictly convex compacta

ON PLI ´ S METRIC ON THE SP A CE OF STRICTL Y CONVEX COMP ACT A MAXIM V. BALASHOV AND DU ˇ SAN R EPO V ˇ S Abstra ct. W e consider a certain metric on the space of all convex compacta in R n , in tro duced by A. Pli ´ s. The set of strictly convex compacta is a complete metric subspace of t he metric space of con vex compacta with respect to this metric. W e present some applications of this metric to th e problems of set-v al ued analysis, in particular w e estimate t he distance b etw een tw o compact sets with resp ect to this metric and to the Hausdorff metric. 1. Introduction W e b egin b y some definitions for a finite-dimensional Euclidean space ( R n , k · k ) o v er R with the inn er pro duct ( · , · ). Let B r ( a ) = { x ∈ R n | k x − a k ≤ r } . Let cl A denote the closur e and int A the interior of the subset A ⊂ R n . The diameter o f the s u bset A ⊂ R n is defined by diam A = sup x,y ∈ A k x − y k . The distanc e from the p oin t x ∈ R n to the s et A ⊂ R n is giv en b y the form ula  ( x, A ) = in f a ∈ A k x − a k . W e shall denote the c onvex hul l of a set A ⊂ R n b y co A . W e shall denote the c onic hul l of a set A ⊂ R n b y cone A (cf. [1, 9, 13]). The Hausdorff distanc e b et w een t w o subsets A, B ⊂ R n is defined as follo ws h ( A, B ) = = m ax  sup a ∈ A inf b ∈ B k a − b k , sup b ∈ B inf a ∈ A k a − b k  = in f { r > 0 | A ⊂ B + B r (0) , B ⊂ A + B r (0) } . The supp orting fu nction of the subset A ⊂ R n is d efined as follo ws s ( p, A ) = s up x ∈ A ( p, x ) , ∀ p ∈ R n . (1 . 1) The supp orting function of a ny set A is alw a ys lo w er s emicon tin uous, p ositiv ely uniform and con v ex. If the set A is b ounded then the supp orting function is Lipschitz cont inuous [1 , 9 ]. It follo ws from the Separation Theorem (cf. [9, Lemma 1.11 .4]) that for an y conv e x co mpacta A, B in R n h ( A, B ) = sup k p k =1 | s ( p, A ) − s ( p, B ) | . (1 . 2) Let ( T ,  ) be a metric s p ace. W e sa y that a set-v alued mapping F : ( T ,  ) → 2 R n \{∅} is upp er semic o ntinuous at the p oin t t = t 0 if ∀ ε > 0 ∃ δ > 0 ∀ t :  ( t, t 0 ) < δ F ( t ) ⊂ F ( t 0 ) + B ε (0) , and lower semic ontinuous at the p oin t t = t 0 if ∀ ε > 0 ∃ δ > 0 ∀ t :  ( t, t 0 ) < δ F ( t 0 ) ⊂ F ( t ) + B ε (0) . W e sa y that a set-v alued mapp ing F : ( T ,  ) → 2 R n \{∅} is c ontinuous at the p oin t t = t 0 if F is upp er and lo w er semicon tin uous at the p oin t t = t 0 . W e say that a set-v alued mappin g F : ( T ,  ) → 2 R n \{∅} is (upp er, lower) (semi)c ontinuous on the set T , if F is (upp er, low er) (semi)co nt inuous at any p oin t t 0 ∈ T . Date : September 26, 2018. 2010 M athematics Subje ct Classific ation. Primary: 54A20, 52A41. S econdary: 52A20, 52A99, 46N10. Key wor ds and phr ases. Metric space, strictly conv ex compactum, mod u lus of conv exity , set-v alued mapping, strict conv exit y , uniform convexit y , su p p orting function, Demy ano v d istance, Hausdorff distance. 1 2 M. V. BALASHOV AND D. REPO V ˇ S F or an y con v ex co mpact set A ⊂ R n and an y vecto r p ∈ R n , the su bset A ( p ) = { x ∈ A | ( p, x ) = s ( p, A ) } is the sub differenti al of the su pp orting fun ction s ( p, A ) a t th e p oint p . Th e set-v alued mapping R n ∋ p → A ( p ) is alw a ys upp er semicont inuous (cf. [1, 13]). A conv e x compactum in R n is called strictly con v ex if its b oundary cont ains no n ondegenerate line segmen ts. D e f i n i t i o n 1.1. ([10]). Let E b e a Banac h space and let a su bset A ⊂ E b e con v ex and closed. The mo dulus of c onvexity δ A : [0 , diam A ) → [0 , + ∞ ) is the fun ction defi n ed by δ A ( ε ) = su p  δ ≥ 0     B δ  x 1 + x 2 2  ⊂ A, ∀ x 1 , x 2 ∈ A : k x 1 − x 2 k = ε  . D e f i n i t i o n 1.2. ([10]). Let E b e a Banac h space and let a su bset A ⊂ E b e con v ex and closed. If the m o dulus of con v exit y δ A ( ε ) is str ictly p ositiv e for all ε ∈ (0 , d iam A ), then w e call the set A uniformly c onvex ( with mo dulus δ A ( · )). F or any un iformly conv e x set A the mo du lus δ A is a strictly increasing f unction on the segmen t [0 , d iam A ). In the finite-dimensional case the class of strictly con v ex compacta coincides with the class of uniformly con v ex compacta with modu li of conv exit y δ A ( ε ) > 0 for all p ermissible ε > 0 (cf. [3 ]). W e shall u se ∗ for ob jects from conjugate space E ∗ : k · k ∗ is the norm in E ∗ , B ∗ 1 (0) is the unit closed ball in E ∗ and so on. F ollo wing Pli ´ s [8] w e define the metric ρ whic h is the main ob j ectiv e of the presen t pap er. D e f i n i t i o n 1.3. ([8, F orm ula (3)]) Th e metric ρ on the sp ace of con v ex compacta in R n is defined b y the form ula ρ ( A, B ) = sup k p k =1 h ( A ( p ) , B ( p )) , (1 . 3) for an y con v ex compacta A, B ⊂ R n . Definition 1.3 coincides with the definition of the Demyanov metric (see its defin ition in [4, F orm u la (4.1)]) – this w as pro v ed in [6]. The Hausdorff metric is th e most natur al metric f or v arious questions of set-v alued an alysis and its applicatio ns. Nev ertheless, there are some limitations for using this metric. F or example, if we ha v e a sequence { A k } ∞ k =1 of strictly conv ex compact sets and h ( A k , A ) → 0, then the limit set A needs not b e strictly conv ex. In deed, consider on the Euclidean p lane the follo wing ellipsoids A k = { ( x 1 , x 2 ) ∈ R 2 | x 2 1 + k 2 x 2 2 ≤ 1 } . Eac h set A k is strictly co nv ex, but the limit set A = { ( x 1 , 0) ∈ R 2 | x 1 ∈ [ − 1 , 1] } is not s trictly con vex. Note, that strict con vexi t y of the set means differen tiabilit y of the sup p orting fu nction of this set. This fact is v ery usefu l f or applications. Belo w we giv e some sufficien t conditions for the limit of a sequence of strictly conv ex compacta to b e also strictly con v ex. W e sa y that a sequence of con vex compacta { A k } ∞ k =1 ⊂ R n is unif ormly c onvex with mo dulus δ if inf k diam A k > 0 and the function δ ( ε ), ε ∈ [0 , inf k diam A k ), is contin uous and has the prop erty 0 < δ ( ε ) ≤ δ A k ( ε ) for all ε ∈ (0 , inf k diam A k ) and f or all k . L e m m a 1.1. L et a se q uenc e { A k } ∞ k =1 ⊂ R n of c onvex c omp acta c onver ge to a c onvex c omp actum A in the Hausdorff metric. If the se quenc e { A k } ∞ k =1 is uniformly c onvex with mo dulus δ , δ : (0 , ε 0 ] → (0 , + ∞ ) , then the c omp actum A is a uniformly c onvex set with the mo dulus δ A ( ε ) ≥ δ ( ε ) , 0 < ε ≤ ε 0 . In p articular, this implies strict c onvexity of the set A . P r o o f. Cho ose arb itrary p oints x, y ∈ A w ith k x − y k < ε 0 . There are tw o sequences { x k } ⊂ A k , { y k } ⊂ A k suc h that x k → x , y k → y , k → ∞ . F or all sufficient ly large k w e ha v e k x k − y k k < ε 0 . Due to the uniform con vexit y of the sequence { A k } we obtain that x k + y k 2 + B δ ( k x k − y k k ) (0) ⊂ A, ON PLI ´ S METRIC ON THE SP ACE OF STRICTL Y CONVEX COMP ACT A 3 and  p, x k + y k 2  + δ ( k x k − y k k ) k p k ≤ s ( p, A k ) , ∀ p ∈ R n . T aking the limit k → ∞ , using (1.2) and the con tinuit y of the function δ we get  p, x + y 2  + δ ( k x − y k ) k p k ≤ s ( p, A ) , ∀ p ∈ R n , i.e. s  p, x + y 2 + B δ ( k x − y k ) (0)  ≤ s ( p, A ) , ∀ p ∈ R n . By the Separation Theorem [9, 13] we obtain the follo wing x + y 2 + B δ ( k x − y k ) (0) ⊂ A.  2. The main proper ties of metric ρ In general, the sub differen tial of a con vex fun ction is only upp er s emicontin uous [1, 13]. F or (n ot strictly) conv ex co mpactum A the sets A ( p ) are also upp er semicont inuous with resp ect to p . This leads to the fact that in the form ula (1.3) from Definition 1.3 one cannot rep lace sup b y max. E x a m p l e 2.1. Consider in R 3 t wo sets: A = co  { ( x 1 , x 2 , x 3 ) | ( x 1 − 1) 2 + x 2 2 = 1; x 3 = 0 } ∪ { (0 , 0 , 1) }  , B = co  { ( x 1 , x 2 , x 3 ) | ( x 1 − 1) 2 + x 2 2 + x 8 2 = 1; x 3 = 0 } ∪ { (0 , 0 , 1) }  . It is easy to see that B ⊂ A , d iam B = diam A = √ 5, and diam A and d iam B are attai ned only on the the line segmen t [(0 , 0 , 1) , (2 , 0 , 0)] ⊂ B . Let a k ∈ { ( x 1 , x 2 , x 3 ) | ( x 1 − 1) 2 + x 2 2 = 1; x 2 < 0; x 3 = 0 } su c h that a k → (2 , 0 , 0). The line segmen t [(0 , 0 , 1) , a k ] is a generatrix of the cone A f or all k . Let H k b e a supp orting plane of the set A such that [(0 , 0 , 1) , a k ] ⊂ H k . Let p k b e a unit normal v ector to th e plane H k suc h that ( p k , a k ) > 0. It is easy to see that p k → p 0 = 1 √ 5 (1 , 0 , 2). F or an y k we h a ve B ( p k ) = { (0 , 0 , 1) } and A ( p k ) = H k ∩ A = [(0 , 0 , 1) , a k ]. By Definition 1.3 it follo ws that ρ ( A, B ) ≥ h ( A ( p k ) , B ( p k )) = h ( { (0 , 0 , 1) } , { (0 , 0 , 1) , a k } ) = k (0 , 0 , 1) − a k k = p k a k k 2 + 1 , and p k a k k 2 + 1 → √ 5, p k a k k 2 + 1 < √ 5 for all k . Ho wev er, the only line segment whic h re- alizes diam A = d iam B = √ 5 is the line segment [(0 , 0 , 1) , (2 , 0 , 0)] ⊂ A ∩ B . Thus ρ ( A, B ) = lim k →∞ h ( A ( p k ) , B ( p k )) = √ 5, but f or all p , k p k = 1, h ( A ( p ) , B ( p )) < √ 5.  L e m m a 2.1. L et A ⊂ R n b e a c onvex c omp actum. If the set-value d mapping B . 1 (0) ∋ p → A ( p ) is lower semic ontinuous, then the c omp actum A is strictly c onvex. P r o o f. S upp ose that th ere exists p ∈ B . 1 (0) suc h that the s et A ( p ) is not a singleton. Let { x, y } ⊂ A ( p ), x 6 = y , and q = y − x k y − x k . Obvio usly , q is orthogonal to p . Consider a sequence { q k } ∞ k =1 ⊂ cone { p, q } suc h that q k → p , k → ∞ , k q k k = 1 and q k 6 = p for all k . Let H − p = { z ∈ R n | ( p, z ) ≤ s ( p, A ) } , H + q k = { z ∈ R n | ( q k , z ) ≥ ( y , q k ) } , H + q = { z ∈ R n | ( q , z ) ≥ ( y , q ) } . By low er semicont inuit y of A ( · ) w e hav e for an y ε > 0 and for all sufficien tly large k A ( p ) ⊂ A ( q k ) + B ε (0) . (2 . 4) On the other hand , A ( q k ) ⊂ H + q k ∩ H − p ⊂ H + q ∩ H − p . (2 . 5) Due to the inclusion (2.5) w e obtain that  ( x, A ( q k )) ≥  ( x, H + q ∩ H − p ) = k x − y k > 0 . (2 . 6) 4 M. V. BALASHOV AND D. REPO V ˇ S Inequalit y (2.6) implies that for all k x / ∈ A ( q k ) + k x − y k 2 B 1 (0) . This con tradicts the inclusion (2.4).  L e m m a 2.2. Consider a se que nc e F k : ( T ,  ) → 2 R n \{∅} of se t- value d mappings which ar e upp er (lower) semic ontinuous with c omp act images. L et the se quenc e { F k ( t ) } ∞ k =1 uniformly c onver ge to the set-value d mapping F : ( T ,  ) → 2 R n \{∅} in the Hausdorff metric, i.e. ∀ ε > 0 ∃ k ε ∀ k > k ε ∀ t ∈ T h ( F k ( t ) , F ( t )) < ε. Then the set-value d mapping F i s upp e r (low er) semic ontinuous on T . P r o o f. The pro of is a standard argumen t of uniform con verge nce.  W e shall write F k ⇒ F , t ∈ T , in the case of uniform con vergence on the set T of the sequence F k to the m ap p ing F . T h e o r e m 2.1. The metric sp ac e of c onvex c omp acta in R n is c omplete with r esp e ct to metric ρ . P r o o f. Let { A k } ∞ k =1 b e a fund amen tal sequence of con v ex compacta with resp ect to metric ρ . This means that ∀ ε > 0 ∃ M ∀ k , m > M ∀ p ∈ B . 1 (0) h ( A m ( p ) , A k ( p )) < ε. By con v exit y of compact sets A m ( p ) and completeness of th e space of con v ex compacta w ith resp ect to th e Hausdorff metric (see [9, Th eorem 1.3.2]) w e obtain that A m ( p ) ⇒ A p , p ∈ B . 1 (0), and the set A p is conv ex and compact for all p ∈ R n , k p k = 1. Put A = cl co [ k p k =1 A p . F or an y q ∈ B . 1 (0) and an y x ( q ) ∈ A q there exists a sequence { x m ( q ) } ∞ m =1 suc h that x m ( q ) ∈ A m ( q ) for all m and x m ( q ) → x ( q ). T aking a limit m → ∞ in the inequalit y ( p, x m ( p )) ≥ ( p, x m ( q )), w e obtain ( p, x ( p )) ≥ ( p, x ( q )). Hence ( p, x ( p )) ≥ s ( p, A ) and x ( p ) ∈ A ( p ), i.e. A p ⊂ A ( p ). The conv erse inclusion A ( p ) ⊂ A p can be pro v ed on the cont rary with the help of separation theorem.  C o r o l l a r y 2.1. The metric subsp ac e of strictly c onvex c omp acta in R n is c omplete with r esp e ct to metric ρ . P r o o f. The pro of is analogous to the pro of of Theorem 2.1 except that all sets A m ( p ), A p are singletons.  Supp ose that A , B are con v ex compacta. By form ula (1.2) we h a ve ρ ( A, B ) = sup k p k =1 sup k q k =1 | s ( q , A ( p )) − s ( q , B ( p )) | , and hence ρ ( A, B ) ≥ sup k p k =1 | s ( p, A ( p )) − s ( p, B ( p )) | = sup k p k =1 | s ( p, A ) − s ( p, B ) | = h ( A, B ) . (2 . 7) Th us ρ ( A k , A ) → 0 implies that h ( A k , A ) → 0. T h e o r e m 2.2. The metric sp ac e of strictly c onvex c omp acta in R n is not lo c al ly c omp act with r esp e ct to the metric ρ . P r o o f. Cho ose a sequence { A k } ∞ k =1 of str ictly con v ex compacta su c h that A k ⊂ B R (0) f or all k and there exists a nonstrictly con vex compact um A with h ( A k , A ) → 0. Supp ose that a su bsequence { A k m } ∞ m =1 con verges to a compactum B in the metric ρ . F rom th e estimate ρ ( A k m , B ) ≥ h ( A k m , B ) and h ( A k m , A ) → 0 w e get equalit y B = A . So ρ ( A k m , A ) → 0. This means that A k m ( p ) ⇒ A ( p ), k p k = 1. But the set A k m ( p ) is a singleton for ON PLI ´ S METRIC ON THE SP ACE OF STRICTL Y CONVEX COMP ACT A 5 all m an d p . By the choice of A there exists p 0 ∈ B . 1 (0) suc h that the set A ( p 0 ) is not a singleton. Con tradiction.  F urther we shall obtain the estimate of distance ρ ( A, B ) via h ( A, B ) for some con ve x closed sets in a Banac h space. In a Banac h space E for closed conv ex b ounded sets A, B ⊂ E we define ρ ( A, B ) = sup k p k ∗ =1 h ( A ( p ) , B ( p )) . If the space E is reflexiv e then A ( p ) 6 = ∅ , B ( p ) 6 = ∅ for all p ∈ E ∗ . Note that if the space E conta ins a u niformly con v ex nonsingleton set then such space E has equiv alen t uniformly con vex norm [3, Theorem 2.3]. In particular, suc h space E is reflexiv e. Note also that for any uniformly con v ex set A w e ha v e that d iam A < + ∞ and the mo dulus of con vexit y δ A ( ε ) is a strictly increasing function w hen ε ∈ [0 , d iam A ) [3]. T h e o r e m 2.3. L et E b e a Banach sp ac e. L et A, B ⊂ E b e c onvex close d b ounde d sets. L et the set A b e a nonsingleton and a uniformly c onvex set with the mo dulus of c onvexity δ A . L et ∆ = lim t → diam A − 0 δ A ( t ) . Then ρ ( A, B ) ≤    h ( A, B ) + δ − 1 A ( h ( A, B )) , h ( A, B ) < ∆ , h ( A, B )  1 + diam A ∆  , h ( A, B ) ≥ ∆ , (2 . 8) wher e the function δ − 1 A is the inverse function to the function δ A . F urthemor e, if the set A is a singleton then ρ ( A, B ) = h ( A, B ) . P r o o f. Let h = h ( A, B ). S upp ose that A is not a singleton. Fix p ∈ B . ∗ 1 (0). Let A ( p ) = { a ( p ) } . Fix an arbitrary p oint b ( p ) ∈ B ( p ). Case 1. h < ∆ . Cho ose t > 1 suc h that th < ∆. Sub case 1.1. s ( p, A ) ≥ s ( p, B ). By formula (1.2) we ha v e 0 ≤ ( p, a ( p )) − ( p, b ( p )) ≤ h . Let a ∈ A b e suc h a p oint that a ∈ b ( p ) + B th (0). Define H A ( p ) = { z ∈ E | ( p, z ) = s ( p, A ) } , H − A ( p ) = { z ∈ E | ( p, z ) ≤ s ( p, A ) } , H B ( p ) = { z ∈ E | ( p, z ) = s ( p, B ) } . W e ha ve  ( b ( p ) , H A ( p )) = ( p, a ( p ) − b ( p )) ≤ h ,  ( a, H A ( p ))) ≤ k a − b ( p ) k +  ( b ( p ) , H A ( p )) ≤ (1 + t ) h and A ∪ B ⊂ H − A ( p ). Hence the line seg ment [ a ( p ) , a ] b elongs to the set H A ( p ) − . Let w = a ( p )+ a 2 ,  ( w , H A ( p )) = 1 2  ( a, H A ( p )) ≤ 1+ t 2 h . By the inclusion w + δ A ( k a ( p ) − a k ) B 1 (0) ⊂ A ⊂ H − A ( p ) w e get δ A ( k a ( p ) − a k ) ≤  ( w , H A ( p )) ≤ 1 + t 2 h. Hence k a ( p ) − a k ≤ δ − 1 A  1+ t 2 h  . Thus we obtain that k a ( p ) − b ( p ) k ≤ k a ( p ) − a k + k a − b ( p ) k ≤ δ − 1 A  1 + t 2 h  + th, i.e. b ( p ) ∈ a ( p ) +  δ − 1 A  1+ t 2 h  + th  B 1 (0). Due to the arbitrary c hoice of the p oin t b ( p ) ∈ B ( p ) w e ha v e B ( p ) ⊂ a ( p ) +  δ − 1 A  1 + t 2 h  + th  B 1 (0) and h ( A ( p ) , B ( p )) = h ( { a ( p ) } , B ( p )) ≤ δ − 1 A  1 + t 2 h  + th. T aking the limit t → 1 + 0, w e obtain th at h ( A ( p ) , B ( p )) = h ( { a ( p ) } , B ( p )) ≤ δ − 1 A ( h ) + h. 6 M. V. BALASHOV AND D. REPO V ˇ S Sub case 1.2. s ( p, A ) < s ( p, B ). T hen all arguments of the sub case 1.1 still apply except that  ( a, H A ( p )) ≤  ( a, H B ( p )) ≤ k a − b ( p ) k ≤ th,  ( w, H A ( p )) ≤ t 2 h , k a ( p ) − a k ≤ δ − 1 A  t 2 h  . Hence h ( A ( p ) , B ( p )) = h ( { a ( p ) } , B ( p )) ≤ δ − 1 A  t 2 h  + th. T aking the limit t → 1 + 0, w e obtain th at h ( A ( p ) , B ( p )) ≤ δ − 1 A  1 2 h  + h. So ag ain when h < ∆ w e ha ve for all p ∈ B . ∗ 1 (0) h ( A ( p ) , B ( p )) ≤ δ − 1 A ( h ) + h. Hence ρ ( A, B ) = sup k p k ∗ =1 h ( A ( p ) , B ( p )) ≤ δ − 1 A ( h ) + h . Case 2. h ≥ ∆ . Then for an y t > 1 w e hav e ρ ( A, B ) ≤ d iam A + th ≤ h ∆ diam A + th ≤ h  t + diam A ∆  , ∀ t > 1 . T aking the limit t → 1 + 0, w e get ρ ( A, B ) ≤ h  1 + diam A ∆  . In the case when A is a singleton the equalit y ρ ( A, B ) = h ( A, B ) follo w s b y definition 1.3 .  F or a set A ⊂ R n , A ⊂ B R ( a ) f or some a ∈ R n and R > 0, we define R- str ongly c onvex hul l of the set A , as th e inte rsection of all closed balls of radius R eac h of which con tains the set A . W e shall denote the R -strongly con v ex hull of the set A by strco R A (cf. [2]). E x a m p l e 2.2. The estimate (2.8) is exact. Consid er t wo sets A and B on the Euclidean plane R 2 . Let 0 < ε < R , a  √ 2 Rε − ε 2 , 0  ∈ R 2 and A = strco R { B ε ((0 , 0)) ∪ { a }} + B R (0) , B = strco R { B ε ( a ) ∪ { (0 , 0) }} + B R (0) . Let p = (0 , 1). It is easy to see th at h ( A, B ) = ε , A ( p ) = ( ε + R ) p , B ( p ) =  √ 2 Rε − ε 2 , ε  + R p . Hence ρ ( A, B ) ≥ h ( A ( p ) , B ( p )) = k a k = p 2 Rε − ε 2 = p 2 Rh ( A, B ) − h 2 ( A, B ) . The sets A and B = a − A are intersec tions o f closed balls of radius R + ε and δ A ( s ) = δ B ( s ) ≥ ( R + ε ) δ H  s R + ε  , where δ H ( s ) = 1 − q 1 − s 2 4 is the mod ulus of conv exit y for the Hilb ert space (see [7, p . 63]). Thus δ A ( s ) = δ B ( s ) ≥ s 2 8( R + ε ) , and δ − 1 A ( t ) ≤ 2 p 2( R + ε ) t . So the order of h ( A, B ) in form u la (2.8) is exact.  R e m a r k 2.1. Th e result of Theorem 2.3 w as pro ve d for p -con v ex sets in [8, F ormula (5)]. Note that any p -conv ex set, p > 0, in the pap er [8] is in fact the in tersection of closed balls of radius R = 1 2 p . F rom the definition of p -conv ex set (inequ alit y (2) of [8]) it follo ws that for an y p -con vex set A ⊂ R n , any p oin t a ∈ A . and an y un it vec tor w ∈ { q ∈ R n | ( q , x − a ) ≤ 0 , ∀ x ∈ A } w e ha v e ( w, x − a ) + p k x − a k 2 ≤ 0 , ∀ x ∈ A, or A ⊂ B R ( a − Rw ) , where R = 1 2 p . Hence A = T k w k =1 B 1 2 p  a ( w ) − 1 2 p w  , w here { a ( w ) } = A ( w ). ON PLI ´ S METRIC ON THE SP ACE OF STRICTL Y CONVEX COMP ACT A 7 This also follo ws by results of [5], [9, Chapter 3]. C o r o l l a r y 2.2. Supp ose that F i : ( T ,  ) → 2 R n \{∅} , i = 1 , 2 , ar e c ontinuous (in the metric ρ ) set-value d mappings with strictly c onvex images. L et L : R n → R n b e a line ar op er ator. Then the set-value d mappings F 1 ( t ) + F 2 ( t ) , LF 1 ( t ) , F 2 ( t ) ∗ F 1 ( t ) = T x ∈ F 1 ( t ) ( F 2 ( t ) − x ) , F 1 ( t ) ∩ F 2 ( t ) (the latter two if nonempty) ar e c ontinuous in the metric ρ . P r o o f. The pro of is similar for all cases. Let us pro v e the co ntin uity of F 1 ( t ) ∩ F 2 ( t ). The conti nuit y of set-v alued mapp ings F i in the metric ρ and formula (2.7) giv es the con tin uit y of set-v alued mappin gs F i in the Hausd orff metric. It is well kn own that the int ersection of t w o con tin uous in the Hausdorff metric set-v alued mappings with compact strictly conv ex images is also con tinuous in th e Hausdorff metric (cf. [1, 9 ]). Th us the set-v alued mappin g H = F 1 ∩ F 2 : ( T ,  ) → 2 R n \{∅} is con tinuous in the Hausdorff metric. F or any p oint t = t 0 ∈ T th e set H ( t 0 ) is a strictly(=un iformly) co nv ex compactum f rom R n with some mo dulus of con v exit y δ t 0 . By Theorem 2.3 w e ha v e ρ ( H ( t ) , H ( t 0 )) ≤ ≤ m ax  h ( H ( t ) , H ( t 0 )) + δ − 1 t 0 ( h ( H ( t ) , H ( t 0 )));  1 + diam H ( t 0 ) ∆  h ( H ( t ) , H ( t 0 ))  → t → t 0 0 , where ∆ = δ t 0 (diam H ( t 0 )). If H ( t 0 ) is a singleton then ρ ( H ( t ) , H ( t 0 )) = h ( H ( t ) , H ( t 0 )) → t → t 0 0.  3. Applica tions 3.1. W e prov e a theorem ab out s mo oth ap p ro ximation of th e extremal pr oblem. T h e o r e m 3.1 . L et F : ( T ,  ) → 2 R n \∅ b e a c ontinuous set- v alue d map ping with c omp act c onvex images and supp ose that ther e e xi sts r > 0 such that f or al l t ∈ T F ( t ) ⊂ B r ( a ( t )) for some a ( t ) ∈ R n . L et diam F ( t ) ≥ d > 0 for al l t ∈ T . F or any t ∈ T and p ∈ R n , k p k = 1 , c onsider the fol lowing pr oblem max { ( p, x ) | x ∈ F ( t ) } . (3 . 9) Then for any ε ∈ (0 , 1) ther e exists an app r oximation F ε : ( T ,  ) → 2 R n \∅ , F ( t ) ⊂ F ε ( t ) f or al l t ∈ T , h ( F ( t ) , F ε ( t )) ≤ ε for al l t ∈ T , such that f or e ach t ∈ T and p ∈ R n , k p k = 1 , the fol lowing pr oblem max { ( p, x ) | x ∈ F ε ( t ) } (3 . 10) has a uniqu e solution F ε ( t, p ) = { f ε ( t, p ) } = arg m ax x ∈ F ε ( t ) ( p, x ) which is H¨ older c ontinuous with th e p ower 1 2 with r esp e ct to h ( F ( t 1 ) , F ( t 2 )) for al l t 1 , t 2 ∈ T . The p ower 1 2 is the b est p ossible in th e gener al c ase. P r o o f. Fix ε ∈ (0 , 1). Let R = max { r 2 ε , r + 1 } . Define F ε ( t ) as the in tersection of all closed balls of radius R , eac h of whic h con tains the set F ( t ). This set is n onempt y b ecause F ( t ) ⊂ B R ( a ( t )). By [2 , formulae (5.7), (5.8)] and [9 , Th eorem 4.4.7 ] we hav e for all t 1 , t 2 ∈ T h ( F ε ( t 1 ) , F ε ( t 2 )) ≤ C ( ε ) h ( F ( t 1 ) , F ( t 2 )) , C ( ε ) = max ( r R + r R − r , 1 + r 2 R ( R − r ) ) . By [2 , Th eorem 5.4] and [9, Th eorem 4.4.6] we hav e h ( F ( t ) , F ε ( t )) ≤ r 2 R ≤ ε, ∀ t ∈ T . By the inequalit y δ F ε ( t ) ( s ) ≥ R δ H  s R  , where δ H ( s ) = 1 − q 1 − s 2 4 is the mo dulus of con ve xit y for the Hilb ert s pace [7, p. 63], w e get δ F ε ( t ) ( s ) ≥ s 2 8 R , ∀ s ∈ (0 , diam F ε ( t )) , 8 M. V. BALASHOV AND D. REPO V ˇ S and b y Theorem 2.3 w e obtain for any p ∈ B . 1 (0) th at k f ε ( t 1 , p ) − f ε ( t 2 , p ) k ≤ ≤ m ax { C ( ε ) h ( F ( t 1 ) , F ( t 2 )) + p 8 RC ( ε ) h ( F ( t 1 ) , F ( t 2 ));  1 + 2 r ∆  h ( F ( t 1 ) , F ( t 2 )) } , where ∆ = R δ H  d R  . On the other h an d , w e ha v e for an y co nv ex compact set A ⊂ R n that f or some constan t C > 0 th e inequalit y δ A ( ε ) ≤ C ε 2 holds for all ε ∈ (0 , diam A ) (see [3]). T aking in to accoun t also Example 2.2, w e see that the p o w er 1 2 is th e b est p ossible.  3.2. W e consider Lipsc hitz selections an d parametrizations of (strictly) conv ex compact sets with metric ρ . With any conv ex compact set A ⊂ R n w e can associate the Steiner p oint s ( A ) = 1 v 1 Z k p k =1 s ( p, A ) p dµ n − 1 , v 1 = µ n B 1 (0) , where µ n is th e Leb esgue measur e in R n . It is w ell known that the Steiner p oint is a Lips c h itz selecti on of con vex compacta in R n with the Hausdorff metric, i.e. for an y con vex compact a A, B ⊂ R n w e h a ve s ( A ) ∈ A and k s ( A ) − s ( B ) k ≤ 2 √ π Γ  n 2 + 1  Γ  n +1 2  h ( A, B ) . The Lipsc hitz constan t (of the order √ n ) ab o v e is the b est p ossible [11]. See also [14, P . 53], [12], [9, Theorem 2.1.2] for details. Using th e Gauss-t yp e form u la (see [9, form ula (2.1 .15)], [12, formula (3.1)]) w e obtain that 1 v 1 Z k p k =1 s ( p, A ) p dµ n − 1 = 1 v 1 Z k p k≤ 1 ∇ s ( p, A ) dµ n . Note that ∇ s ( p, A ) exists a.e. on the ball B 1 (0). F or an y con v ex compactum A ⊂ R n define U ( A ) = { p ∈ B 1 (0) | ∃ ∇ s ( p, A ) } . T h e function s ( p, A ) is Lips chitz contin uous h ence µ n U ( A ) = µ n B 1 (0). Let a ( A, p ) = ∇ s ( p, A ) for p ∈ U ( A ) and a ( A, p ) = 0 for p ∈ B 1 (0) \ U ( A ). Let A, B ⊂ R n b e conv ex compacta and U = U ( A ) ∩ U ( B ), µ n U = µ n B 1 (0). Then k s ( A ) − s ( B ) k ≤ 1 v 1 Z U k a ( A, p ) − a ( B , p ) k dµ n ≤ 1 v 1 Z U ρ ( A, B ) dµ n = ρ ( A, B ) . Th us the Steiner p oint is a Lipsc hitz selection of con v ex compacta in R n with metric ρ with the Lipsc hitz constan t 1. Let A b e a collecti on of strictly conv ex compacta. Th en for an y p ∈ R n , k p k = 1, the function a ( p ) = A ( p ), A ∈ A , is a Lipsc hitz selection of the family A with th e L ipsc hitz constan t 1 in th e metric ρ . T h e o r e m 3.2. L e t a c ol le ction of strictly c onvex c omp acta A fr om R n b e uniformly b ounde d, i.e. ther e exists M > 0 such that k A k = h ( { 0 } , A ) ≤ M for al l A ∈ A . Then ther e exists the f amily of fu nctions f λ,p : A → R n , ( λ, p ) ∈ [0 , 1] × B . 1 (0) , (3 . 11) such that for any A ∈ A we have A = { f λ,p ( A ) | λ ∈ [0 , 1] , p ∈ B . 1 (0) } and for any ( λ, p ) ∈ [0 , 1] × B . 1 (0) the fu nction f λ,p is Lipschitz on A ∈ A sele c tion in the metric ρ with Lipschitz c onstant 1. Mor e over, the f unction [0 , 1] × B . 1 (0) ∋ ( λ, p ) → f λ,p ( A ) is c ontinuous for any A ∈ A and the function f λ,p is additive: f λ,p ( A + B ) = f λ,p ( A ) + f λ,p ( B ) , A, B ∈ A . ON PLI ´ S METRIC ON THE SP ACE OF STRICTL Y CONVEX COMP ACT A 9 P r o o f. F or any A ∈ A we define f λ,p ( A ) = λa ( p ) + (1 − λ ) s ( A ) ∈ A . Let A, B ∈ A . Then (note, that b ( p ) = B ( p ) for an y p ∈ B . 1 (0)) k f λ,p ( A ) − f λ,p ( B ) k ≤ λ k a ( p ) − b ( p ) k + (1 − λ ) k s ( A ) − s ( B ) k ≤ ≤ λρ ( A, B ) + (1 − λ ) ρ ( A, B ) = ρ ( A, B ) . Cho ose λ 1 , λ 2 ∈ [0 , 1] and p 1 , p 2 ∈ B . 1 (0) an d A ∈ A . k f λ 1 ,p 1 ( A ) − f λ 2 ,p 2 ( A ) k = k λ 1 a ( p 1 ) + (1 − λ ) s ( A ) − λ 2 a ( p 2 ) − (1 − λ 2 ) s ( A ) k ≤ ≤ k λ 1 a ( p 1 ) − λ 2 a ( p 2 ) k + | λ 1 − λ 2 |k s ( A ) k ≤ | λ 1 − λ 2 |k a ( p 1 ) k + | λ 2 |k a ( p 1 ) − a ( p 2 ) k + + | λ 1 − λ 2 |k s ( A ) k ≤ 2 | λ 1 − λ 2 | M + k a ( p 1 ) − a ( p 2 ) k . The gradien t a ( p ) = ∇ s ( p, A ) for the strictly conv ex compact set A is u niformly con tinuous on the unit sphere (see [3, Lemma 2.2]). S o the fu nction [0 , 1] × B . 1 (0) ∋ ( λ, p ) → f λ,p ( A ) is un iformly con tinuous. By the Moreau-Ro c k afellar theorem [13] for all A, B ∈ A we get A ( p ) + B ( p ) = ( A + B )( p ) for all p ∈ B . 1 (0). Using the additiv e pr op ert y of the Steiner p oin t [9], [12], [14] w e obtain that f λ,p is an additiv e selection for all λ ∈ [0 , 1] and k p k = 1.  R e m a r k 3.1. W e see from the pro of of Theorem 3.2, that the function ( λ, p ) → f λ,p ( A ) is u n iformly contin uous for any A ∈ A . More precisely , f λ,p is Lip s c h itz on λ ∈ [0 , 1] (with Lipschitz constan t 2 M ) and uniformly con tin uous on p ∈ B . 1 (0). Note that f λ,p ( A ) is Lipsc hitz on p ∈ B . 1 (0) if and only if the set A is an in tersection of closed balls of the same fixed radius. The last assertion follo ws b y results of [5] and by Theorem 4.3.2 of [9 ]: a set A is the interse ction of close d b al ls of fixe d r adius R > 0 in Hilb ert sp ac e if and only if k a ( p ) − a ( q ) k ≤ R k p − q k for al l p, q ∈ B . 1 (0) . Here a ( p ) = A ( p ). A cknowledgements This researc h wa s supp orted by SRA gran ts P1-0292-010 1, J1-2057 -0101, and BI-R U/10-11/ 002. The first author w as sup p orted b y RFBR gran t 10-01-00 139-a, ADAP pro ject ”Dev elopment of scien tific p oten tial of higher sc ho ol” 2.1.1/1 1133 and pro jects of F AP ”Kadr y” 1.2.1 grant P938 and gran t 16.740.1 1.0128. W e thank the referee for comment s and suggestions. Referen ces [1] J.-P. Aub in, I. Ekeland, Ap plied Nonlinear An alysis, John Wiley & Sons Inc., New Y ork, 1984. [2] M. V. Balashov, E. S. Polovinkin, M -strongly conv ex subsets and t h eir generating sets, Sb ornik: Math. 191:1 (2000), 25-60. [3] M. V . Balashov, D. R ep ov ˇ s, Uniform con vexit y and the spliting problem for selec tions, J. Math. Anal. Appl. 360:1 ( 2009), 307-316. [4] P. Diamond, P. Kl o e den, A. R ubinov, A. Vladimir ov, Comparative p roperties of three metrics in t he space of compact conv ex sets, Set- V alued Anal. 5:3 (1997), 267-289. [5] H. F r ankowska, Ch. 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Prz eslawski, Linear and Lipsc h itz contin u ou s selectors fo r t h e family of conv ex sets in Euclidean v ector spaces, Bull. Pol ish Acad. Sci. Math. 33:1-2 (1985), 31–34. [12] K. Przeslawski, Lipsc hitz contin uous selectors, Part I: Linear selectors, J. Conv ex Anal. 5:2 (1998), 249-267. [13] R.T. Ro ckafel lar, Conv ex Analysis, Princeton Universit y Press, Princeton, NJ, 1970. 10 M. V. BALASHOV AND D. REPO V ˇ S [14] R. Schneider, Convex Bodies: The Brunn- Minko wski Theory , Cambridge Univ. Press, 1993. Dep ar tmen t of High er Ma thema tics, Moscow Institute of Physics and Technology, Institutski str. 9, Dolgopr udny, Mosco w region, Russia 141700. balashov@mail.mipt.ru F acul ty of Ma thema tics and Physics, and F acul ty of Educa tion, Uni versity of Ljubljana, Jadran- ska 19, Ljubljana, Slovenia 1000. dusan.repovs@guest.arnes.si