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
.