NiraDynandElzaFarkhi. The application of spline subdivision schemes to data consisting of convex compact sets,

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1 Splne Subdvson Schemes for Convex Compact Sets NraDynandElzaFarkh The applcaton of splne subdvson schemes to data consstng of convex compact sets, wth addton replaced by Mnkowsk sums of sets, s nvestgated. These methods generate n the lmt set-valued functons, whch can be expressed explctly n terms of lnear combnatons of nteger shfts of B-splnes wth the ntal data as coecents. The subdvson technques are used to conclude that these lmt set-valued splne functons have shape preservng propertes smlar to those of the usual splne functons. Ths extenson of subdvson methods from the scalar settng to the set-valued case has applcaton n the approxmate reconstructon of 3-D bodes from nte collectons of ther parallel cross-sectons. Keywords: convex sets, support functons, Mnkowsk addton, set-valued functons, splne subdvson, shape preservaton, approxmaton. 1 Introducton Subdvson schemes are recursve methods for the generaton of smooth functons from dscrete data. By these methods at each recurson step, new dscrete values on a ner grd are computed by weghted sums of the already exstng dscrete values. In the lmt of the recursve process, data s dened on a dense set of ponts. Consderng ths data as functon values, under certan condtons, a lmt contnuous functon s dened by ths process. The theory of subdvson processes s presented n e.g. [4], [6]. In ths work we apply a class of subdvson methods wth postve weghts to data consstng of convex compact sets, replacng the addton by Mnkowsk sums of sets, and obtan n the lmt set-valued functons. Ths extenson of subdvson methods from the scalar settng to the set-valued case has applcaton n the approxmate reconstructon of 3-D bodes from nte collectons of ther parallel cross-sectons. The class of subdvson methods consdered here conssts of methods whch generate splne functons n the scalar settng. These methods have shape preservng propertes and approxmaton propertes n the set-valued case, smlar to those n the scalar case. The proof of the shape preservng propertes of the lmt set-valued functons reles on the subdvson technque. Yet the lmt multfuncton (set-valued functon) has a smple explct form n terms of the ntal sets and nteger shfts of B-splnes, and there s no need to compute t recursvely. Ths explct form also yelds the smoothness and the approxmaton propertes of the lmt set-valued functon. Among the splne subdvson schemes, there s a scheme whch generates n the lmt pecewse lnear nterpolants to the gven data of sets. In [11] pecewse lnear approxmaton to convex set-valued functons s studed. Ths approxmaton conssts of a pecewswe lnear nterpolant to samples of a multfuncton, wth addton replaced by Mnkowsk sums. Thus, one can regard the present paper as an extenson of [11]. 1

2 The mathematcal tools used for analysng set-valued functons nclude the support functon technque for descrbng convex compact sets (see e.g. [10]) and methods of embeddng the cone of convex compact subsets of R n n a lnear normed space, wth addton dened as the Mnkowsk sum of sets ([2], [7], [8], [9]). In any such lnear normed space, we ntroduce a partal order generated by the set ncluson order n the cone of convex compact sets. Thus amultfuncton wth convex compact mages from R to R n s consdered as an abstract functon wth values n a partally ordered normed lnear space. Monotoncty and convexty of such abstract functons are easly expressed by the postvty of ther rst and second nte derences. The paper s organsed as follows. In secton 2 basc facts about convex compact sets, ther support functons and ther embeddng n a normed lnear space wth a partal order, are presented. Secton 3 presents a smple example of a shape-preservng set-valued subdvson scheme. The man results about splne subdvson methods appled to convex compact sets are derved n Secton 4. The explct form of the lmt multvalued splne functon s obtaned together wth ts approxmaton and shape-preservng propertes. 2 Prelmnares Denote by C(R n ) the cone of nonempty convex compact subsets of R n : Recall the dentons of Mnkowsk sum and multplcaton by scalars of sets A B 2C(R n ): A + B = f a + b j a 2 A b 2 B g A = f a j a 2 A g: Snce the technque of support functons s central to ths text, we recall the denton and the basc propertes of these functons (see e.g. [7], [10]). For a set A 2C(R n ) ts support functon (A ) :R n! R s dened as follows: (A l) = maxhl a l a2a 2 Rn where h s the Eucldean nner product. Note that for any xedl 2 R n (A l) s nte. The followng propertes of are well known ([10]): 1 o : (A l) = (A l) 0: 2 o : (A l 1 + l 2 ) (A l 1 )+ (A l 2 ): 3 o : (A ) s Lpschtz contnuous: wth a constant kak = max kak where kk s the a2a Eucldean norm, namely j (A l 1 ) ; (A l 2 )jkakkl 1 ; l 2 k,forl 1 l 2 2 R n. 4 o : A scalar functon : R n! R s a support functon of a convex compact set t satses 1 o 2 o (see e.g. [10], Theorem 13.2 and ts corollares). 5 o : (A + B ) = (A )+ (B ): 6 o : (A ) = (A ) 0 7 o : A B () (A l) (B l) for each l 2 R n : 2

3 8 o : The Hausdor dstance between two sets A B 2C(R n ) s gven, n terms of the support functons of these two sets, by where S n;1 s the unt sphere n R n : haus(a B) = max j (A l) ; (B l)j l2s n;1 Note that by 1 o and 2 o, (A ) s postvely homogeneous and convex. Also, functons satsfyng 1 o,2 o are called sublnear. As we noted n 3 o, (see e.g. [10]), sublnear functons dened on all R n are Lpschtz contnuous. The next proposton wll be used n what follows. Proposton 2.1 Let A 1 A 2 B 1 B 2 2C(R n ) and A 1 B 1 : Then the equalty A 1 + B 2 = A 2 + B 1 mples A 2 B 2 : Proof: The proof follows from propertes 5 o 7 o : A 1 + B 2 = A 2 + B 1 () (A 1 )+ (B 2 ) = (A 2 )+ (B 1 ): From A 1 B 1 follows (A 1 ) (B 1 ): Ths, combned wth the above equalty mples (B 2 ) (A 2 ) =) B 2 A 2 : There are varous ways to construct a lnear normed vector space D(R n ) relatve to the Mnkowsk sum, n whch the cone C(R n )sembedded by anembeddng J : C(R n )! D(R n ) wth the followng propertes(see e.g. [7], [9], [8]): () J(A + B) =J(A)+J(B): () J(A) =J(A) 0: () J(A) =J(B) () A = B: (v) kj(a) ; J(B)k =haus(a B). These propertes mply J(f0g) =, where s the zero element ofd(r n ). A smple embeddng s J(A) = (A ). It s easy to check, usng the above stated propertes of support functons, that ths embeddng has the requred four propertes. Havng an embeddng of the cone C(R n )nto a lnear normed vector space D(R n ), we can ntroduce a partal order n D(R n ) and therefore n C(R n ). For that, we dene the followng cone n D(R n ): K = n C 2D(R n )j C = J(A) ; J(B) A B2C(R n ) A B The cone K determnes the followng partal order n D(R n ): For A B 2D(R n ) A B () B ; A 2K: (2) Remark 2.1 Here are some observatons regardng the partal order ntroduced above: 1. By () and () K s a convex cone. o (1) 3

4 2. Note that (1) does not depend on the choce of the sets A and B n the followng sense: f C = J(A 1 ) ; J(B 1 )=J(A 2 ) ; J(B 2 ) where A 1 B 1 then by () and () we get A 1 + B 2 = A 2 + B 1 and by Proposton 2.1 A 2 B 2 : 3. By the prevous observaton, the order dened nd(r n ) by (2) nduces the regular ncluson order n C(R n ) namely for A B 2C(R n ) A B () J(A) J(B): Thus we denote for A B 2C(R n ) A B A B. 4. Our denton of the postve cone K n D(R n ) concdes wth the postve cone n some concrete lnear spaces n whch C(R n ) s embedded, lke the space ofthepars of convex compact sets and the space of derences of support functons (see e.g. [8]). Remark 2.2 It follows from the facts that K s a cone and D(R n ) s a lnear space that for A B (a) ;A ;B (b) A + C B + C for every C 2D(R n ) (c) If C D, then A + C B + D for 0: The prevous remark justes the notons of postve and negatve elements of D(R n ). The element A 2D(R n )scallednonnegatve when A.e.A 2K: If A A 6= then A s called postve. The element B 2D(R n )scallednonpostve f ;B s nonnegatve and B s negatve when ;B s postve. We wll call a convex compact set A nonnegatve when J(A),.e. 0 2 A: A s postve 0 2 A and A 6= f0g. We are nterested n ths work n set-valued mappngs from R to C(R n ) (called also multmaps or multfunctons), and n partcular n multmaps of the form F (t) = N =1 A f (t) (3) where A 2C(R n ) f : R 7! R f (t) 0forallt 2 R. LetS be the cone of maps of the form (3). We saythatf 2Ss C k f n (3) f 2 C k for =1 ::: N. Denton 2.1 A mappng F : R!D(R n ) s called (A) monotone ncreasng f t 1 t 2 mples F (t 1 ) F (t 2 ) (.e. F (t 2 ) ; F (t 1 ) 2K). (B) monotone decreasng f F (;t) s monotone ncreasng. (C) convex f F (t 1 +(1; )t 2 ) F (t 1 )+(1; )F (t 2 ) for each 2 [0 1] t 1 t 2 2 R: (4) 4

5 (D) concave f F (;t) s convex. Denton 2.2 Dene for a gven functon F : R!D(R n ) the k-th forward nte derence at the pont t wth a step h>0 Forasequence ff g we dene k h F (t) = k j=0 ( k F ) = (;1) k;j k j k j=0! (;1) k;j k j F (t + jh):! F +j : In case 1 F s nonnegatve for all t h > 0 (), the functon (sequence) s monotone ncreasng. If 2 F s nonpostve for all t h > 0 (), the functon (sequence) s called convex. Remark 2.3 For the sake of smplcty we sometmes dentfy the set A 2 C(R n ) wth ts embedded mage J(A) and for sets A B 2C(R n ) we denote by A ; B the derence J(A) ; J(B). Geometrcally, a monotone ncreasng set-valued map F : R!C(R n ) has a growng mage as the argument t ncreases. Formally, F s monotone when for a gven h>0 the rst derence 1 h (t) =F (t + h) ; F (t) s of a constant sgn. Smlarly, the convexty means that the second derence 2 hf (t) =F (t)+f(t +2h) ; 2F (t + h) s nonpostve (.e. F (t)+f(t +2h) 2F (t + h)). The nequalty (4) s opposte to the common denton of convex scalar functons. We choose t ths way n order to ensure the convexty of the graph of F: Clearly, n the case of a convex map F : R!C(R n ) F (t 1 +(1; )t 2 ) F (t 1 )+(1; )F (t 2 ) for each 2 [0 1] t 1 t 2 2 R whch means that the graph of F s a convex set n R n+1 : The last ncluson and property 7 o of support functons mply that for each gven drecton l the support functon (F () l) of the convex multmap F s a concave scalar functon. 3 Chakn Subdvson Scheme for Convex Compact Sets Let F 0 =0 ::: N be convex compact sets n Rn : We seek a set-valued functon n S whch has a smlar structure to the the pecewse lnear multfuncton F : R! R n dened by F ( + ) =(1; )F 0 + F =0 1 ::: N ; 1 but s smoother n the sense of (3). Note that every pecewse lnear multfuncton s n S and s C 0. 5

6 Consder the followng teratve procedure of reconstructng F known as Chakn algorthm when appled to scalar functons ([5], [6]). At level k +1 k 0 of the procedure we calculate for =0 ::: 2 k (N ; 1) the sets 2 = 1 4 F k F k (5) 2 = 3 4 F k F k : (6) Thanks to the postvty of the coecents, we obtan convex compact sets at each stage of the process. Moreover, snce the coecents form a convex combnaton, the scheme has two notceable propertes: 1. It preserves monotoncty,.e. If for all F k ;1 F k then 2;1 2 2 for all. Ths follows drectly from the equaltes 2 ; 2;1 = 1 4 (F k ; F;1) k 2 ; 2 = 1 2 (F k ; F k ) (7) whch means that the rst nte derences reman n K at each stage of the process f they are n K for k =0: 2. It preserves convexty, namely the second derences reman n K at each teraton f they are n t for k = 0, snce ; 2 2 = 1 4 (F k + F k +2 ; 2F) k 2;1 + 2 ; 2 2 = 1 4 (F k ;1 + F k ; 2F k ): Atthek-th teraton (k 1) we construct a pecewse lnear multmap F k :[0 N;1+2 ;k ]! C(R n ) satsfyng F k (t) = tk ; t t k ; t k ;1 F k ;1 + t ; tk ;1 t k ; t k F k tk ;1 t t k : ;1 Lemma 3.1 The sequence of set-valued functons ff k (t)g 1 k=1 converges (unformly n the nterval [0 N ; 1]) to a Lpschtz contnuous multfuncton F 1 (t) wth convex values. Proof: Denote the support functons k (l) = (F k l)andk (l t) = (F k (t) l): Then by (5), (6) and propertes 5 o 6 o of support functons t follows that for each xed drecton l 2 R n k+1 2 (l) = 1 4 k (l)+ 3 4 k (l) k+1 2 (l) =3 4 k (l)+1 4 k (l): 6

7 It means that for each gven l the Chakn subdvson procedure s realzed on the scalar values k (l). Hence by the well known convergence of the Chakn algorthm for scalar subdvson (see e.g. [6]) for each xedl2r n there exsts a lmt functon 1 (l t) = lm k!1 k (l t): In the followng we prove that 1 (l t) s Lpschtz contnuous n l and t. Let us x a pont t 2 [0 N ; 1]: By propertes 1 o 2 o of the support functons k ( t), t follows that the lmt functon 1 ( t) satses 1 o 2 o. Moreover, snce the ntal sets F 0 k =0 :::N are unformly bounded, then by (5),(6)F =0 :::2k (N ; 1) + 1 and F k (t) are unformly bounded by the same constant. Hence by property 3 o the functons k ( t) are Lpschtz contnuous wth an absolute constant, ndependent of t and k, and therefore the lmt functon 1 ( t)slpschtz contnuous wth the same constant and the convergence s unform wth respect to l. Moreover, propertes 1 o 2 o characterzng each support functon hold for the lmt functon 1 ( t) for xed t: Therefore, by 4 o 1 ( t) s the support functon of a convex compact set whch sdenotedby F 1 (t): That F 1 (t) = lm F k (t) can be concluded from k!1 haus(f 1 (t) F k (t)) = max l2s 1 j 1 (l t) ; k (l t)j!0 as k!1: To see that ths convergence s unform for t 2 [0 N ; 1], we observeby (7) and by property (v) of J that haus(f k Fk ) ;1 2;k max haus(f 0 j+1 F0 j ): 0jN;1 Ths means that the Lpschtz constants of the pecewse lnear maps F k (t), and therefore of F 1 (t), ( or of the functons k (l ) 1 (l ) ) do not exceed the constant max haus(f 0 1N F0 ;1): Hence k (l ) andf k () are unformly Lpschtz contnuous wth the same constant nthe nterval [0 N;1] and converge unformly to 1 (l )andf 1 ()) respectvely, onthsnterval. The shape preservng propertes of F 1 (t) and ts smoothness follow from the analyss of more general schemes, done n the next secton. 4 A Class of Shape Preservng Subdvson Schemes Let the ntal sequence F 0 =0 ::: N of convex compact sets n Rn be gven. For convenence we dene F 0 = f0g for 2 Zn f0 1 ::: Ng. Consder a ntely supported subdvson scheme gven by = a [m] ;2j F k j 2 Z k =0 1 2 ::: (8) wth the splne weghts a [m] m+1 = =2 m = 0 1 ::: m+ 1 and a [m] f0 1 ::: m +1g: Note that Chakn algorthm s the specal case m =2. = 0 for 2 Zn 7

8 The scheme (8) when appled to scalar values ff 0 g, s unformly convergent and ts lmt functon f 1 () s of the form (see e.g. [6]) f 1 (t) = 2Z f 0 B m(t ; ) (9) where the functon B m () s a B-splne of degree m, wth nteger knots and support [0 m+1]: In the followng we obtan a set-valued analog of (9). As n the prevous secton, at the k-th teraton (k 1) we construct a pecewse lnear multmap F k : R!C(R n ) satsfyng F k (t k )=F k for t k =2 ;k : Let the generatng functon of the sequence fa [m] g be a [m] (z) = m+1 =0 a [m] z : Then a [m] () hastheform Clearly the coecents a [m] a [m] (z) = (1 + z)m+1 2 m : are nonnegatve numbers and satsfy 2Z a [m] 2 =1 2Z a [m] 2 =1: (10) Denote the derences G k = J(F k ) ; J(F k ;1 ) 1 k 0: Note that Gk 2D(Rn ). It s easy to show as n the case of scalar functons (see e.g.[6]) that Proposton 4.1 The derences G k satsfy G k+1 = j b [m] ;2j Gk j 2 Z k =0 1 ::: (11) where the generatng functon b [m] () of the sequence fb [m] g s b [m] (z) = 2Z b [m] z = a[m] (z) 1+z = (1 + z)m 2 m : (12) It s clear from the last Proposton that the coecents b [m] are nonnegatve and [ m 2 ] =0 [ m b [m] 2 ] 2 = =0 b [m] 2 = 1 2 : (13) Ths means that f the ntal sets ff 0 g N =0 form a monotone ncreasng sequence (.e. the derences G 0 =1 ::: N are n the cone K, then the derences Gk =1 ::: N reman n ths cone at each stage of the subdvson process (11),.e. for each k the sequence ff k g s monotone ncreasng. 8

9 By the same reasonng the second derences H k = G k ; G k ;1 can be obtaned by a subdvson scheme wth a generatng functon c [m] (z) = b[m] (z) 1+z = (1 + z)m;1 2 m : (14) Snce c [m] (z) has nonnegatve coecents, the second derences reman n K at each teraton k, provded they are n K for k = 0. Therefore the scheme (8) s shape preservng. By the above argumentaton t s easy to conclude that the subdvson schemes (8) preserve the sgn of the derences F k of order 1 m.e. ( F k ) belongs to K, provded ( F 0 ) 2K for all. Thus we have proved the followng Proposton 4.2 The subdvson scheme (8) s monotoncty and convexty preservng,.e. F k j F k j+1 for all j =) F k+1 for all (15) F k + F k 2F k j;1 j+1 j for all j =) + ;1 k+1 2F + for all : (16) Theorem 4.1 The set-valued mappngs ff k ()g 1 k=1 multmap F 1 () wth convex mages of the form converge unformly on R to a splne F 1 (t) = 2Z F 0 B m(t ; ) for each t 2 R: (17) Proof: Fx a drecton l 2 R n and denote k (l) = (F k l), k (l t) = (F k (t) l). Then (8) mples k+1 (l) = a [m] ;2j k j (l) hence by (9) for each l 1 (l t) = 2Z 0 (l)b m (t ; ): Snce the last sum s nte for any t, and snce the coecents B m (t ; ) are nonnegatve, t follows that for xed t the scalar functon 1 ( t) s a support functon of the set F 1 (t) = 2Z F 0 B m(t ; ): (18) The fact that F 1 (t) = lm k!1 F k (t) can be proved by the same argumentaton as n the proof of Lemma 3.1 wth (7) replaced by (11) and (13). The next corollary follows from the fact that B m 2 C m;1. Corollary 4.1 F 1 2S and F 1 2 C m;1. For m =1, F 1 s pecewse lnear multfuncton nterpolatng ff 0 g.e. satsfyng F 1 () = F 0 F1 (t) =(t ; )F 0 0 +( +1; t)f for t +1 2 Z. For m = 2 (the Chakn scheme), F 1 2 C 1 and t s pecewse quadratc n the sense that for each xedl the support functon (l ) s a lnear combnaton of quadratc B-splnes. The followng shape preservng propertes of F 1 follow from Proposton 4.2 and the dscusson above t. 9

10 Corollary 4.2 Let ff 0 g N =1 be an ntal sequence ofcompact convex sets. (a) If the ntal sequence ff 0 g N =1 s monotone ncreasng,.e. F 0 F 0 =1 ::: N; 1, then the map F 1 (t) s monotone ncreasng n the sense of Remark 2.3. (b) If the ntal sequence sconvex,.e. F 0 + F 0 F 0 +2 =0 :::N; 2, then the map F 1 (t) has a convex graph. Proof: (a) Wth the notatons of the prevous theorem, t follows from (15) that for every l 2 R n k (l) k (l): Snce a pecewse lnear nterpolatng scalar functon of monotone data s tself monotone, t follows that for each l 2 R n k (l t + h) k (l t) for every t and h>0. The last nequalty mples F k (t + h) F k (t) for all t and h>0. Therefore the set-valued map F k () s monotone ncreasng. The last concluson holds also for the lmt F 1 (t) = lm k!1 F k (t). The proof of (b) s smlar, based on (16) and the fact that a scalar pecewse lnear nterpolant of convex (concave) data s convex (concave). Note that n ths case the setvalued functons are convex, but ther support functons are concave for each xed drecton l (see Remark 2.3). The lmt multmaps generated by the above subdvson processes approxmate smooth multmaps n S wth the same approxmaton order as the lmt functons generated by the correspondng scalar schemes approxmate smooth scalar functons. Proposton 4.3 Let F 2S\C r wth r>0. Then haus(f (t) F (jh)b m ( t h ; j)) chs where s = mnfr 2g and the constant c depends only on F. Proof: Let F (t)= P A f (t), where I s a nte subset of Z, andf 2 C r. Denote 2I ~f (t) = P f (jh)b m ( t ; j). Then by awell-known approxmaton result for scalar functons h (see e.g. [3]), we obtan kf ; f ~ k 1 c h s : P wth the constant c dependng only on f.let~ F(t) = A f ~ (t), then 2I ~F(t) = B m ( t ; j)f (jh): h 10

11 Hence for each t haus(f (t) F(t)) ~ = kj( A f (t)) ; J( A ~ f (t))k ka kkf ; f ~ k 1 ch s (19) 2I 2I 2I where we used property (v) of the embeddng J. For general multmaps (not from S) whch are Hausdor contnuous, we get a weaker approxmaton result. Proposton 4.4 Let F be a general multmap wth values n C(R n ) whch s Hausdor contnuous, then haus(f (t) F (jh)b m ( t ; j)) = o(1): h Proof: Foraxedt there s only a nte number of terms n P F (jh)b m ( t ; j) : Now h snce P P B m ( t ; j) =1 we can wrte F (t) = F (t)b h m ( t ; j). Usng an argument smlar h to (19) and by the fact that F s contnuous, we obtan haus(f (t) F (jh)b m ( t h ; j)) = k j2( h t ;m h t )\Z J(F (t)) ; J(F (jh)) B m ( t ; j)k = o(1): h If we assume enough smoothness on the support functons of the sets F (t) for all t,we get a smlar result to that of Proposton 4.3. Proposton 4.5 Let F be a multmap dened onr wth values n C(R n ) such that the t-dependent support functon (F () l) hasasecond dervatve n t unformly bounded nt and n l 2 S n;1. Then haus(f (t) F (jh)b m ( t h ; j)) = O(h2 ): Then Proof: Let us denote the splne multfuncton S h m F (t) = Denote for any l 2 R n F (jh)b m ( t h ; j) = j2( t h ;m t h )\Z F (jh)b m ( t h ; j): haus(f (t) (S h m F )(t)) = k (F (t) ) ; ((S h m F )(t) )k 1 S n;1 : l (t) =(t l)= (F (t) l). For scalar functons t s known that jf(t) ; (S h f)(t)j 1 m 2 sup j d2 f(t) jh 2 : (20) t dt 2 Set f(t) = l (t). Snce Sm h (F () l)= (Sm h F () l), we get (F (t) l) ; ((S h m F )(t) l)j 1 2 k d2 11 dt 2 l()k 1 h 2 :

12 Now, by our assumptons sup l2s n;1 j d2 dt 2 l ()j L, andwe obtan for every t k (F (t) ) ; ((S h m F )(t) )k 1 S n;1 1 2 Lh2 : Note that f F s dened n the nte nterval [0 N], then the estmate near the boundary of the nterval s O(h). Ths follows from the correspondng result n the scalar case. Before concludng the paper, we state a conjecture that was nspred by the famous theorem of R. Aumann [1] statng that the ntegral of a compact-valued multmap s a convex set. Snce the Remann ntegral s the lmt of Remann sums, whch are n essence averages wth postve weghts, we expect that the repeated applcaton of (8) wth the splne weghts generates n the lmt a convex- valued multfuncton, even when the ntal sets are compact but not convex. Conjecture: Any of the subdvson methods (8) wth m>1 appled to ntal compact sets, ffj 0 g generates as a lmt a convex-valued multfuncton, gven by F 1 (t) = cof 0 j B m(t ; j) where coa denotes the convex hull of A. References [1] R. J. Aumann,Integrals of set-valued functons, J. of Math. Analyss and Applcatons, 12 (1965) 1{12. [2] R. Baer and E. Farkh,Drected Sets and Derences of Convex Compact Sets, n: M.P. Pols, A.L. Dontchev, P. Kall, I. Lasecka and A.W. Olbrot, eds., Systems Modellng and Optmzaton, Proc. of the 18th IFIP TC7 Conference, CRC Research Notes n Mathematcs, (Chapman and Hall,1999) 135{143. [3] C. deboor,a Practcal Gude to Splnes (Sprnger Verlag, New York,1978). [4] A. S. Cavaretta, W. Dahmen and C.A. Mcchell,Statonary Subdvson, Memors of AMS, No. 453 (1991). [5] G. M. Chakn,An algorthm for hgh speed curve generaton, Computer Graphcs and Image Processng, 3 (1974) 346{349. [6] N. Dyn,Subdvson schemes n Computer-Aded Geometrc Desgn, n: W. Lght, ed., Advances n Numercal Analyss, Vol. II, Wavelets, Subdvson Algorthms and Radal Bass Functons (Clarendon Press, Oxford, 1992) 36{104. [7] L. Hormander,Sur la foncton d'appu des ensembles convexes dans un espace localement convexe, Arkv for Matematk, 3 (1954)181{

13 [8] B. Margols,Compact, Convex Sets n R n and a new Banach Lattce, I.- Theory, Numer. Funct. Anal. Optmz. 11 (1990) 555{576. [9] H. Radstrom,An embeddng theorem for spaces of convex sets, Proceedngs of the Amercan Mathematcal Socety 3 (1952) 165{169. [10] R. T. Rockafellar,Convex Analyss (Prnceton Unversty Press, Prnceton, 1970). [11] R. A. Vtale,Approxmatons of convex set-valued functons, J. Approx. Theory 26 (1979) 301{316. Nra Dyn nradyn@math.tau.ac.l Elza Farkh elza@math.tau.ac.l School of Mathematcal Scences Sackler Faculty of Exact Scences Tel Avv Unversty, Tel Avv, Israel 13