Best approximation in metric spaces

Pub . Ma . UAB Vol . 27 N2 2 Juny 1983 BEST APPROXIMATION IN METRIC SPACES T . D . Na ang * 1 . In oduc ion . The no iono s ic con exi y and uni- o m con exi y in no medlinea spaces was ex ended o me ic spaces in [1] and ce ain exis ence and uniqueness heo ems on bes app oxima ion we e p o ed in hese spaces in [1] and [2] . In his no ewe shall gi e a ela ionship be ween he wo ypes o con exi ies in me ic spaces and u he discuss some esul s on bes app oxima ion in me ic spaces . We shall also ex end he no ion o sun in oduced in no med linea spaces by E imo and S eckin [31 o me ic spaces . 2 . S ic ly - Con ex And Uni o mly - Con ex Me ic Spaces . I x,y,z a eany hesepoin s in a me ic space  (X,d) hen  z  is said o be a poín be ween  x and y  i d(x,z) + d(z,y) = d(x,y) . * The au ho is hank ul o he U .G .C . o inancial suppo . 7 1 Fu he , z is said o be a m¿d-po .ín o  x and y i d(x,z) = d(z,y) = 2 d(x,y) . A me ic d de ined on X  is said o be a con ex me A .í,c i o eachpai x,y in X, d de e mines a leas one mid- poin and d  is said o be a  s &ongly con ex me h .¿c i o e e y pai i de e mines a unique mid-poin . A me ic space  (X,d)  is said o be a con ex me lcíe epace o a  s Aongly con ex me iJc .i .e epace  acco ding as he me ic  d  is con exo s ongly con ex . A s onglycon ex me ic space (X,d) is said o be exh,íc X .y con ex  i d(x,x o ) < , d(y,x 0 ) < imply d(z,x 0 ) < , unless x = y, whe e x o is a bi a y bu . ixed noin o X, z is he mid-poin o x and y, and is any ini e eal numbe . A s onglycon exme icspace (X,d) is said o be un¿bo,únly con ex  i he e co esponda o each pai o posi i e numbe s  (E, ) a posi i e numbe S such ha d(x,y) <e whene e d(x,x o ) < , d(y,x ) < , d(z,x 0 ) > - 8,  z being he mid-poin o x and y and he o he poin sbeing a bi a y . 7 2 A me ic space  (X,d)  is said o be  o aily comple e  i e e y bounded closed subse o X is compac . As is easy o see, a compac space is o ally comple e bu a o ally comple espaceneed no be compac e .g . he eal line wi h he usual me ic . I was shown in [1] ha e e yuni o mly con ex me ic space is s ic ly con ex and a compac s ic ly con exme ic space is uni o mly con ex . Howe e , we ha e : Theo em 1 . E e y o allycomple e s ic ly con ex me ic space is uni o mly con ex . he se The p oo gi en in [1] wo ksin his si ua ion oo as S = { G x,y > : d(x,x o ) -< , d(y,X O ) -< , d(X,y) > e } is closed as well as bounded in he o allycomple espace XxX and so compac . 3 . Bes App oxima ion Ma In a Me icSpace . Gi en a subse  K  o a me ic space (X,d)  and  xE X, a poin  yOE K  such ha  d (x,y o )  = d (x,K)  is called a  po .ín og bes appnox .íma c :on  o  x  in  K . The mapping k which akeseachpoin o he space o hosepoin so K which a e nea es o i , is called he me x íc pico jec c :on . We shall deno eby K (x) hese o bes app oxima ion elemen s o x in K i .e . The se K is said o be puxim¿nal i each poin o X has a bes app oxima ion in K and i is said o be Chebyehe i each poin o X has a unique bes app oxima ion in K . Theo em 2 . I G is a Chebyshe subse o a me ic space  (X,d)  hen  G (z)  = i  (x)  whe e  z EX  is any elemen be ween x and Y G (x) . implies K (x)  =  {kEK  :  d(x,k)  = d(x,K)} . Fo Chebyshe se s he me ic p ojec ion is single- alued . P oo . By he de ini ion o z Le  g E G .  Then = d(z,i G(x)) d(x,Z) + d(z,i G (x) ) = d(x,u G(x)) . d(x,z) + d(z,g) > d(x,g) d(z,g) > d(x,g) - d(x,z) > d(x,i G (x)) - d(x,z) i . e .  d(z, G (x))  < d(z,g)  o all  gEG  and so  7T G (x) EG  is a bes app oxima ion o  zE X .  Since  G  is Chebyshe , 7 G (x)  = n G (z) . Rema k 1 . This esul is analogous o he ollowing esul p o ed in no med linea spacesby M . Nicolescu (see Lemma 2 .1[4], p . 364) Le E be a no med linea space and G a Chebyshe se in E, hen G [ax + (1 - a) G (x)1 = G (x), xEE, 0 < a< 1 . Rema k 2 . The concep o a 'sun' was in oduced in app oxima ion heo y by E imo and S eekin [3] as : A Chebyshe subse G o a no med linea space E is called a sun i we ha e i G [ax  + (1-  cx)7T G (x)1  =  i G (x),  xEE,  a  > 0 i .e .  i  w G (x) EG  is bes app oxima ion o  x E E  hen  G (x) is also a bes app oxima ion o all poin s on he ay G (x)~ As is easy o see, his concep is meaning ul in a y linea me- ic space . Mo i a ed by Theo em 2, we now ex end he no ion o sun o any me ic space (X,d) . A poin  z E X  is said o be on he  nay  xy  i ei he z is be ween x and y o y is be ween x and z i .e . ei he d(x,y) = d(x,z) + d(z,y) o d(x,z) = d(x,y) + d(y,z) . A Chebyshe se G in a me ic space (X,d) is called a  zun  i o each  xE X, i G (z) = nG (x)  o e e y  z  on he ay  i G (X)k . I will be in e es ing o s udy suns in me ic spaces . Theo em 3 . I (X,d) is a me ic space, G a subse o X  and  goE G,  hen  G1 (go)  = {x E X  :  d (x,g o )  = d (x,G)}  is clo sed and  x o Ei -1(go) =>  zE nG l ( g o )  o e e y  z  be ween  x o and g . 0 P oo . nG 1 (g o ) _ {xEX : d(x,go ) = d(x,G)} = {xEX : d(x,g 0 ) < d(x,g) o all gEG} = gnG {xEX : d(x,g o )  d(x,g)} . The closedness o i G 1 (g 0 )  now ollows om he con inui y o  d . Now  xo E i G 1 (g o )  d(xo ,g o ) 5 d(xo ,g)  o all  g E G . Since  z  is be ween  xo and  go , d(x o ,z) + d(z,g 0 ) = d(xo,go) . W i e d(z,g) > d(g,x 0 )  - d(x o ,z)  o all  g E G d(x o ,g o ) - d(x0,z) = d(z,g0) i .e .  d(z,g 0 )  S d(z,g)  o all  gEG i .e .  z E i G1 (go) . Rema k 1 . This esul is analogous o he ollowing e- sul p o ed in no med linea spaces (see [4] . p .143 and p .354) : Le X be a no med linea space, G a linea subspace o  X  and  g oEG .  Then he se  i G l (g o )  is closed and x E G1 (go)  F ax  + (1 - a)  go E G1 (g o ),  0 <- a 5 1 . Rema k 2 . I G is a Chebyshe se in X hen Theo em 3 gi es ha  wG1 (7 G (x) )  is closed o e e y  xEX . I is a mapping om a non-emp y se X in o a non-emp y se Y hen he gnaph o is he se G( ) = {(x, (x)) : XEX} . I is wellknown ha o con inuous mancnings in me ic 77 spaces he g aph is closed . I is also well known ha o Che- byshe se s he me ic p ojec ion need no necessa ily be con i- nuous . Howe e , we ha e : Theo em 4 . I K is a Chebyshe se in a me ic space (X,d) hen he g aph o he me ic p ojec ion a K is closed . This implies . y --> Yand n P oo .  G( K ) = { (x, 7T K (x)) : XEX} . Le  (y, z) exis s a sequence  < (ynpi K(yn)) > in G(i K )  such ha Conside be a limi poin o G( K ). Then he e (Y n I K (Y n ) )  -  (y . z) . I d (Y n ,7 K (Y n ) )  -  d (Y 'l K (Y)) (  =  I d (Y n .K)  -  d (Y .K) I d(Y n ,Y) . So, d(yn, K(yn)) < d(Yn 1Y) + d(Y .n K (Y))  implies d(Y,z) <- d(Y,Yn ) + d(Y n .7 K (y n )) + d(7 K(yn)'z) 2d (Y .Y n )  + d(y,7 K (Y))  + d(7 K(Yn)'z) . Thisimplies d(Y,z) -~ d(Y,7K(Y)) . Since  K  is closed,  z EK  and so  d(y,z) % d(y,nK(y)) .  Thus d(y,z) = d(y,nK(y)) . Since K is Chebyshe , z = uK (y) i .e . (y, z) EG (n K )  and so  G (n K )  is closed .