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a
Lipsc hitz continuity of best approximations and Chebyshev
centers
LI chong' and WANG xinghua2
1 Ce211erlor M a t h e r t i a r t c ~ Sclcnce*,
l
2he)lang Uni\ersity. Hangzhou 310027, C h l n a ; 2 . Department of Mathematics. Hs~lgzhou
Ln~rer+rty.Hangzhnu 3111028. China
Ahrlrad
The L r p ~ c h ~CDI:IIILU~~J.
tt
t e s d ~ 3~
Ipm P r p , c ~ r ~ ~by
e dSCPI::CICI and VjrtL
IS
r given
r
for b e ~ appr0xirnd~i0n5
t
and
Chebphev
centers. Consequently.
the open prob-
w l v e d completely.
h s t spprmlmation. Chcbyshev center. untlorm mnvexlty.
Kryu'nrds
a subset in Bannrll space X, ond -4 be a bounded subset in X . An eiement g o € G 1s called a
restricted Cl~ebysl~ev
center of A with respect to G ~f sup,$
II a - go II = r, ( A ), where I-(; { A ) =
~ n f , +csup, E ,4 11 a - g 11 is the restricted Cliebyshcv radius of A with respect to G . ?-he set af all such go
is denoted by PG( A ) . In particular, if A is a singleton r. as is known, go is called a hest appmxinlation
ro x from G . Furthermore, u-e write r G ( A )= r ( A ) and PC;(.4) = P( A ) for G = 1.
The study of the Chebysher center problem, which can bc dated back to at least 1962' ' I , has been
deeply investigatedF2'3' and applied in different areas of mathematics such as continuous complexity and
set-maps theory. On the continuity of Chebysl~evcenters, Theorem A was prored in reference [ a ] .
Theorem A . Let X be a Hilbert space. Thcn for any rompact subsets A and 8 ,there l ~ a l d ~
LET C; be
-
<
( r ( A )+ r ( B ) h ( A . B ) ) h ( - 4 , I o ,
11 P ( A ) - P ( B ) 11
(0.1)
~2Ilereh ( A , B 1 is the Hausdorff metric of A and B .
Also they presented two open problems as follows.
Problem 1 k e s ( 0 . 1 ) remain valid without the compact assumprion?
Problem 2 . Is there a sharp estimate of ]I P ( A ) - P ( B ) [I similar to ( 0 . 1 ) In e uniformly convex
Banach space?
Problem 1 hx been solved in the affirmative by one of the authors'". For Problem 2 , Wang and
s'u~'' gave the following estimate.
Theorem B. Let X he n p-unifomdy cnnves Bni~achspace. Then there exists a constant c >0 such
.
that
<
11 P ( A ) - P ( B ) 11 * c ( ( a + h ( A , B ) ) ' - p,
(0.2)
rI-41, r ( B ) l , j3=nlin/ r ( A ) , r ( E ) 1 .
L'nfortunately, rullen X is a Hilbert space ( 0 . 2 ) does riot imply (0.1) because of constant c . Thus
nacurai to ask whether there existi. a betrer estlmatc such that ( 0 . 2 ) gives ( 0 . 1 ) for Hilbert space
wljrre a =rnaxl
lr is
X.
I n the prrsent narc w e Iirsf i n r r o d ~ ~ ct he e concept of p-uniformly convexity on D with respect to G
and give an esrirnnre inr operator PG(x ) of besr nppnlutnatton. Semndlv, we use the estimate to solve
t h e a h w e problem cunlplrt
+.
I Lipschitz continuit) of b s l approximation
Ler D be a convex rone. and
1
8 ( ~=
) i n f ,l
-
Then, for p 2 2 ,
Chinese
j: .I,
X
+
-.
il
xm
i .?aid
~
Science Bulletin
to
C;
hv
;
l
I ~ n r t l rsuhjpacc ~n X
T E D, ilrr-yil
=
be p- uniformly convex on D
Vol .43 N3.3
.
E.
For O < s < 2 define
.r-yf
G,
~ v i t respect
l ~
to
February 1998
IIsIi = IIyII
= I
.
G if there exists cp >O such
185
that
d?( E )
>c$ . Obviously,
when,
G = D = X. it is just the p-uniform
Let
(1.1)
It follows from ref. [ 7 1 that X is p-uniformly convex on D with respect to G if and only ii ti, >0.
Now we are ready to prorre the main theorem of this section.
Theorem 1 . 3 . Suppose that
is a clclsed convex subset in X and X is p-uniformly convex on D
wit11 respect t o spanG. Then for any 6 , _v f D there holds
(1.2)
Proof. By induction, it folluws from ( 1 . I ) that
for a:ly
I,
y € D,
;t
--
y
E spanG
and any n = 1, 2 , 3 , ..- . Hence we 11;1vc
(1.3)
This implies
The pr09f
15
con~pletrd.
Remark 1 . l . .'Utlio~g!iTheorem 1. 1
study oi UleLpllev centers (see section 2 ) .
2 Lipschitz continuity of Chehj'sbev
is
verv h r e c t , it xvould be very helplu1 for applicatlot~to the
centers
&
Let X be n p-uniformly convex Bnnacll space, and
he the value oI cl, in ( 1 . 1 ) by taking D = G
D be the set of all bounded convex clmed s u l s e ~ sin X,endowed ivirli the I4ausdorff rnct-
= X'. .Ur)w let
186
Chinese Science Bullefin
Vol 43 No. 3
February 1998
ricf-" . Define operations in D as ioIlows:
A + B = ! a + 6: u E A , b E B t , ;cA = ] l a : a E A1
for any A , B f D and A >0. Then by ref. [ 81 , D can be embedded in a Bnnacll space ( E ,
such that for any A , B E I3
h ( A , B ) = I\ A - B II ,
and D is a convex cone in E under the above operattons.
Clearly, E is p-uniformly convex on D with respect 10 X and
11 11
E)
-
>dp.
O k m e PG( A ) = PG( COA1,
) . Tllus, using Theorem 1.1, we immediately obtain
r G ( A = rG( COA
Theorenl 2 . 1 .
Theorem 2.1. Assume that li is a p-uniformly convex Banach space. Let G be a dosed convex subset in X . T l ~ e nfor any b u n d e d subset A , B in X , there holds
1
The folkawing theorenl shnws that the order - of 14 Fc ( A ) - P, (B)11 is best possible.
0
Theorem 2 . 2 . 1-et p 2 2 . If there exists a constant d >0 such that for any A , B D
I[ P ( A ) - P ( B ) I[
d h (co.4, mB),
then .Y iis p-uniforn~lyconvex.
<
Froor. F o r a n r , ~ > O a n d a n y x , > l f X ' w i t h
IIJII
=
1
1
!I y I I
T11tr1 P ! A j =I), P ( B ) = -(s- y ) and h ( A , B ) = l -7
4
-
1
4'J
This irnplle~r!l*t 6 ( E ) 2-E'
If X is a Hilbert space,
for p<2['"
s+y
I1
[IZ-~II
= E ,
let
. It follows from ( 2 . 2 ) that
. Tl~eproof is completed.
X
1
is 2-uniformly convex an? d : = 4 '
uniformly convex for p 2 2 and #<2,
dl>,
I!-
=1,
(2.2)
If X is the L,-spaces, X is p - and 2-
respecti\.ely.F u r t h e m u r e . in t l l i s case, d , =
. Thus using Theorem 2.1.
w e have
,
1
2
for
>3[q1 and
the follr,i-:\rlg cornllar~es.
Corollary 2 . 1 . Suppose that X is a Hilbert sp3ce a n d t h a t C I F a clrhierl convex subwt it]
for any bounded subsets -4, 6 in X,
11 P c ( A ) - P ( ; ( B ) 11
( r G ( A )+ r I ; ( R )+ h(ro.4. r o B ) ) h ( c o A . cc7E).
Corollary 2 . 2 . Let X be the L,-spaces, 1< p < + K ,, and L; be a clo.;rd convex subser l a
for any bounded s u b s ~ t sA , E in X ,
<
Itemark 2 . 1 . Let us recall thar in a Hilbert space t h e r e lluld=['
<
x.Thcn
17.3)
x . Then
'
(2.3)
1-r Y I I . f c , l - a r ~ ? . > .: i b f X Note Tileoren 2 . 2 . It is interesting to compare ( 2 . 4 ) rvit 11 ( 2 . 3 ) , wlljrh sllo.xs r h o r t h e r e i~ an evenrial
I P I )
-
P
disti~ctionIxtwcer, approxin~ntionsand Cllebysher centcrb.
Chinese Science Bulletin
VoI, 4 3 No. 3
February 1998
187
This work was jointly supported by China Major Key Project Ior Basic Researcher and Nationd N a l u r a l k i e n c e
Foundation of China, partly supported by Zhejimg Provincial Naturd Science Foundation, and China Postdoctoral Science FoundaAcknowledgement
Lion.
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Chinese Science Bulletin
Vol .43No. 3
February 1998