Best approximation in metric spaces

Pub . Mat . UAB Vol . 27 N2 2 Juny 1983 BEST APPROXIMATION IN METRIC SPACES T . D . Narang * 1 . Introduction . The notionof strict convexity and uniform convexity in normedlinear spaces was extended to metric spaces in [1] and certain existence and uniqueness theorems on best approximation were proved in these spaces in [1] and [2] . In this notewe shall give a relationship between thetwo types of convexities in metric spaces and further discuss some results on best approximation in metric spaces . We shall also extend the notion of sun introduced in normed linear spaces by Efimov and Steckin [31 to metric spaces . 2 . Strictly - Convex And Uniformly - Convex Metr ic Spaces . If x,y,z areany thesepoints in a metric space  (X,d) then  z  is said to be a poínt between  x and y  if d(x,z) + d(z,y) = d(x,y) . * The author is thankful to the U .G .C . for financial support . 7 1 Further, z is said to be a m¿d-po .ínt of  x and y if d(x,z) = d(z,y) = 2 d(x,y) . A metric d defined on X  is said to be a convex metA .í,c if for eachpair x,y in X, d determines at least one midpoint and d  is said to be a  st&ongly convex meth .¿c if for every pair it determines a unique mid-point . A metric space  (X,d)  is said to be a convex metlcíe epace or a  stAongly convex mefiJc .i .e epace  according as the metric  d  is convexor strongly convex . A stronglyconvex metric space (X,d) is said to be exh,íctX .y convex  if d(x,x o ) < r, d(y,x 0 ) < r imply d(z,x 0 ) < r, unless x = y, where x o is arbitrary but . fixed noint of X, z is the mid-point of x and y, and r is any finite real number . A stronglyconvexmetricspace (X,d) is said to be un¿bo,únly convex  if there corresponda to each pair of positive numbers  (E,r) a positive number S such that d(x,y) <e whenever d(x,x o ) < r, d(y,x ) < r, d(z,x 0 ) > r - 8,  z being the mid-point of x and y and the otherpointsbeing arbitrary . 7 2 A metric space  (X,d)  is said to be  totaily complete  if every bounded closed subset of X is compact . As is easy to see, a compact space is totally complete but a totally completespaceneed not be compacte .g . the real line with the usual metric . It was shown in [1] that everyuniformly convex metric space is strictly convex and a compact strictly convexmetric space is uniformly convex . However, we have : Theorem 1 . Everytotallycomplete strictly convex metric space is uniformly convex . the set The proof given in [1] worksin this situation too as S = { G x,y > : d(x,x o ) -< r, d(y,X O ) -< r, d(X,y) > e } is closed as well as bounded in the totallycompletespace XxX and so compact . 3 . Best Approximation Ma In a MetricSpace . Given a subset  K  of a metric space (X,d)  and  xE X, a point  yOE K  such that  d (x,y o )  = d (x,K)  is called a  po .ínt og bestappnox .íma tc :on  to  x  in  K . The mapping rk whichtakeseachpointof the space to thosepointsof K which are nearestto it, is called the metx íc pico jec tc :on . We shall denoteby r K (x) theset of best approximation elements of x in K i .e . The set K is said to be puxim¿nal if each point of X has a best approximation in K and it is said to be Chebyehev if each point of X has a unique best approximation in K . Theorem 2 . If G is a Chebyshev subset of a metric space  (X,d)  then  G (z)  = ir  (x)  where  z EX  is any element between x and Y G (x) . implies K (x)  =  {kEK  :  d(x,k)  = d(x,K)} . For Chebyshev sets the metric projection is single-valued . Proof . By the definition of z Let  g E G .  Then = d(z,irG(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,ir G (x)) - d(x,z) i . e .  d(z,rr G (x))  < d(z,g)  for all  gEG  and so  7T G (x) EG  is a best approximation to  zE X .  Since  G  is Chebyshev, 7rG (x)  = n G (z) . Remark 1 . This result is analogous to the following result proved in normed linear spacesby M . Nicolescu (see Lemma 2 .1[4], p . 364) Let E be a normed linear space and G a Chebyshev set in E, then rG [ax + (1 - a)r G (x)1 = rr G (x), xEE, 0 < a< 1 . Remark 2 . The concept of a 'sun' was introduced in approximation theory by Efimov and Steekin [3] as : A Chebyshev subset G of a normed linear space E is called a sun if we have ir G [ax  + (1-  cx)7T G (x)1  =  ir G (x),  xEE,  a  > 0 i .e .  if  w G (x) EG  is best approximation to  x E E  then  rr G (x) is also a best approximation to all points on theray rr G (x)~ As is easy to see, this concept is meaningful in ary linear metric space . Motivated by Theorem 2, we now extend the notion of sun to any metric space (X,d) . A point  z E X  is said to be on the  nay  xy  if either z is between x and y or y is between x and z i .e . either d(x,y) = d(x,z) + d(z,y) or d(x,z) = d(x,y) + d(y,z) . A Chebyshev set G in a metric space (X,d) is called a  zun  if for each  xE X, ir G (z) = nG (x)  for every  z  on the ray  ir G (X)k . It will be interesting to study suns in metric spaces . Theorem 3 . If (X,d) is a metric space, G a subset of X  and  goE G,  then  rG1 (go)  = {x E X  :  d (x,g o )  = d (x,G)}  is clo sed and  x o Eir-1(go) =>  zE nG l ( g o )  for every  z  between  x o and g . 0 Proof . nG 1 (g o ) _ {xEX : d(x,go ) = d(x,G)} = {xEX : d(x,g 0 ) < d(x,g) for all gEG} = gnG {xEX : d(x,g o )  d(x,g)} . The closedness of ir G 1 (g 0 )  now follows from the continuity of  d . Now  xo E ir G 1 (g o )  d(xo ,g o ) 5 d(xo ,g)  for all  g E G . Since  z  is between  xo and  go , d(x o ,z) + d(z,g 0 ) = d(xo,go) . Write d(z,g) > d(g,x 0 )  - d(x o ,z)  for 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)  for all  gEG i .e .  z E ir G1 (go) . Remark 1 . This result is analogous to the following result proved in normed linear spaces (see [4] . p .143 and p .354) : Let X be a normed linear space, G a linearsubspace of  X  and  g oEG .  Then the set  ir G l (g o )  is closed and x E rrG1 (go)  F ax  + (1 - a)  go E rG1 (g o ),  0 <- a 5 1 . Remark 2 . If G is a Chebyshev set in X then Theorem 3 givesthat  wG1 (7r G (x) )  is closed for every  xEX . If f is a mapping from a non-empty set X into a non-empty set Y then the gnaph of f is the set G(f) = {(x, f(x)) : XEX} . It is wellknown that for continuous mancnings in metric 77 spaces the graph is closed . It is also well known that for Chebyshev sets the metric projection need not necessarily be continuous . However, we have : Theorem 4 . If K is a Chebyshev set in a metric space (X,d) then the graph of the metric projection a K is closed . This implies . y --> Yand n Proof .  G(rr K ) = { (x, 7T K (x)) : XEX} . Let  (y, z) exists a sequence  < (ynpirK(yn)) > in G(ir K )  such that Consider be a limit point of G(rr K ). Then there (Y n Ir K (Y n ) )  -  (y . z) . I d (Y n ,7 rK (Y n ) )  -  d (Y 'lr K (Y)) (  =  I d (Y n .K)  -  d (Y .K) I d(Y n ,Y) . So, d(yn,rrK(yn)) < d(Yn 1Y) + d(Y .n K (Y))  implies d(Y,z) <- d(Y,Yn ) + d(Y n .7r K (y n )) + d(7rK(yn)'z) 2d (Y .Y n )  + d(y,7t K (Y))  + d(7rK(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 Chebyshev, z = uK (y) i .e . (y, z) EG (n K )  and so  G (n K )  is closed .