ANNALES
UNIVERSITATIS MARIAE CURIE-SKŁODOWSKA
L U B L I N – P O L O N I A
VOL. LIX, 2005 SECTIO A 9–17
RAFA ESP´
INOLA
On selec ions o he me ic p ojec ion
and bes p oximi y pai s in hype con ex spaces
Dedica ed o W. A. Ki k on he occasion o
his ecei ing an Hono a y Doc o a e om
Ma ia Cu ie-Skłodowska Uni e si y
Abs ac . In his wo k we p esen new esul s on nonexpansi e e ac ions
and bes p oximi y pai s in hype con ex me ic spaces. We sha pen he main
esul s o R. Esp´ınola e al. in [3] (Nonexpansi e e ac s in hype con ex spaces,
J. Ma h. Anal. Appl. 251 (2000), 557–570) on exis ence o nonexpansi e se-
lec ions o he me ic p ojec ion. Mo e p ecisely we cha ac e ize hose subse s
o a hype con ex me ic space wi h he p ope y ha he me ic p ojec ion
on o hem admi s a nonexpansi e selec ion as a subclass o se s in oduced in
[3]. This is a a he excep ional p ope y wi h a lo o applica ions in app ox-
ima ion heo y, in pa icula we apply i o answe in he posi i e he main
ques ion posed by Ki k e al. in [5] (P oximinal e ac s and bes p oximi y
pai heo ems, Num. Func . Anal. Op . 24 (2003), 851–862).
1. In oduc ion. In [3] he au ho e al. in oduced a subclass o sub-
se s o a me ic space, he so-called weakly ex e nally hype con ex subse s
2000 Ma hema ics Subjec Classi ica ion. 47H09, 41A65, 46B20.
Key wo ds and ph ases. Hype con ex spaces, me ic p ojec ions, p oximinal se s, bes
p oximi y pai s.
The wo k o he au ho is pa ially suppo ed by he Minis y o Science and Technology
o Spain G an BFM 2003-3893-C02-01 and Jun a de Andaluc´ıa p ojec FQM127.
10 R. Esp´ınola
(see Sec ion 2 o de ini ions), wi h he goal o cha ac e izing hose sub-
se s o a me ically con ex me ic space o which he e exis s a nonexpan-
si e selec ion o he me ic p ojec ion. In his wo k we gi e he de ini i e
solu ion o he main p oblems s udied in ha pape . Mo e p ecisely i
is p o ed ha i Mis a me ically con ex me ic space, Ais a weakly
ex e nally hype con ex o Mand PAis he me ic p ojec ion on A(i.e.
PA(x) = {y∈A:d(x, y) = in {d(x, u) : u∈A}}) hen he e exis s a
nonexpansi e selec ion Ro PA, his is
R(x)∈PA(x)and d(R(x), R(y)) ≤d(x, y) o x, y ∈M.
This is a a he excep ional p ope y o a subse o a me ic space which has
a la ge numbe o nice consequences ela ed o bes app oxima ion esul s
(see o ins ance [3] and e e ences he ein). We apply his esul o answe
in he posi i e a ques ion on bes p oximi y pai s posed by Ki k e al.
in [5]. Bes p oximi y pai s aise in a e y na u al way in app oxima ion
heo y when s udying he p oximi y o wo se s. Fo a p ope mo i a ion
on bes p oximi y pai s and hei ela ion o ixed poin heo y he eade
may check [5] and e e ences he ein. A subse Eo a me ic space Mis
said o be p oximinal i gi en any x∈M he e exis s px∈Esuch ha
d(x, px) = dis (x, E) = in {d(x, y) : y∈E}. Fo Aand Bnonemp y
subse s o a me ic space le dis (A, B) = in {d(x, y) : x∈Aand y∈B}.
De ini ion 1.1. Le Xbe a me ic space and le Aand Bbe nonemp y
subse s o X. Le
A0={x∈A:d(x, y) = dis (A, B) o some y∈B};
B0={x∈B:d(x, y) = dis (A, B) o some y∈A}.
A pai (x, y)∈A0×B0 o which d(x, y) = dis (A, B)is called a bes
p oximi y pai o Aand B.
In pa icula , i is p o ed in [5] ha i Mis a hype con ex me ic space
and Aand Ba e nonemp y admissible subse s o M, hen A0and B0a e
nonemp y and hype con ex (see also P oposi ion 2.14 in [5]). As a conse-
quence o ou main heo em we can ex end his esul o Aand Bweakly
ex e nally hype con ex subse s o M. Nex his is applied o answe in he
posi i e a ques ion posed in [5]. The las esul o his wo k is ano he
applica ion o ou main heo em, in his case we ob ain he nonexpansi e
e sion o he Ky Fan’s heo em gi en in [3] o hype con ex spaces.
2. De ini ions and p elimina y esul s. This sec ion con ains he de i-
ni ions and esul s ha will be needed in he sequel. Hype con ex me ic
spaces we e in oduced in 1956 by A onszajn and Pani chpakdi in [1], o a
de ailed exposi ion on hype con ex spaces he eade may consul he ecen
su ey on hem by he au ho and Khamsi [2].
On selec ions o he me ic p ojec ion... 11
De ini ion 2.1. A me ic space Mis said o be hype con ex i gi en any
amily {xα}o poin s o Mand any amily { α}o eal numbe s sa is ying
d(xα, xβ)≤ α+ β
i is he case ha TαB(xα; α)6=∅.
Nex we gi e he de ini ion o wo subclasses o subse s o me ic spaces.
De ini ion 2.2. A subse Ao a me ic space Mis said o be admissible
(in M) i i is an in e sec ion o closed balls o M. Thus Ais admissible i
A=Ti∈IB(xi, i)whe e xi∈Mand i≥0 o i∈I.
De ini ion 2.3. A subse Eo a me ic space Mis said o be ex e nally
hype con ex ( ela i e o M) i gi en any amily {xα}o poin s in Mand
any amily { α}o eal numbe s sa is ying
d(xα, xβ)≤ α+ βand dis (xα, E)≤ α
i ollows ha TαB(xα; α)∩E6=∅.
Ex e nally hype con ex subse s we e shown in [4] o enjoy nice p ope ies
as, o ins ance, being always p oximinal. The ollowing heo em also gi es
a e y impo an p ope y o ex e nally hype con ex se s.
Theo em 2.4 ([4]).Le Mbe hype con ex, Sa me ic space and T?a
mul i alued mapping om Sin o Msuch ha T?(x)is bounded nonemp y
ex e nally hype con ex o each x∈S, hen he e exis s a selec ion T:S→
Mo Tsuch ha :
d(T(x), T(y)) ≤dH(T?(x), T?(y)) o all x, y ∈S,
whe e dHdeno es he usual Hausdo me ic on he amily o nonemp y
bounded closed subse s o M.
The ollowing no ion plays a c ucial ole in his wo k.
De ini ion 2.5. A subse Eo a me ic space Mis a p oximinal nonex-
pansi e e ac o Mi he e exis s a nonexpansi e e ac ion Ro Mon o
E o which
d(x, R(x)) = dis (x, E)
o each x∈M. Thus d(R(x), R(y)) ≤d(x, y) o each x, y ∈M.
In an e o o cha ac e ize hose subse s o a hype con ex me ic space
which a e p oximinal nonexpansi e e ac s, he ollowing de ini ion was
in oduced in [3].
De ini ion 2.6. A subse Eo a me ic space Mis said o be weakly ex-
e nally hype con ex ( ela i e o M) i Eis ex e nally hype con ex ela i e
o E∪ {z} o each z∈M. P ecisely, gi en any amily {xα}o poin s in M
12 R. Esp´ınola
all bu a mos one o which lies in E, and any amily { α}o eal numbe s
sa is ying
d(xα, xβ)≤ α+ β,wi h dis (xα, E)≤ αi xα/∈E,
i ollows ha TαB(xα; α)∩E6=∅.
I di ec ly ollows om he de ini ion ha weakly ex e nally hype con ex
subse s a e p oximinal. A his poin i is in e es ing o no e ha when he
h ee classes o subse s so a p esen ed a e subse s o he same hype con ex
me ic space M, hen hey a e ela ed in he ollowing way: le Abe a subse
o M, hen
Ais admissible (in M)⇒Ais ex e nally hype con ex ( ela i e o M)
⇒Ais weakly ex e nally hype con ex ( ela i e o M)
⇒Ais hype con ex.
The nex de ini ion was in oduced in [6].
De ini ion 2.7. Le Abe a subse o a me ic space M. A mapping R:
A→Mis said o be ε-cons an i d(x, R(x)) ≤ε o each x∈A.
Fo Aas abo e he ε-neighbo hood o Ais de ined as ollows:
Nε(A) = [
a∈A
B(a, ε).
The ollowing ac will be needed.
Lemma 2.8 ([3]).Le Abe a weakly ex e nally hype con ex subse o a
hype con ex me ic space M, hen o any ε > 0 he se Nε(A)is weakly
ex e nally hype con ex and he e is an ε-cons an nonexpansi e e ac ion
o Nε(A)on A.
3. P oximinal nonexpansi e e ac s and bes p oximi y pai s. We
begin his sec ion by ecalling Theo em 3.1 in [3].
Theo em 3.1. Suppose Ais a weakly ex e nally hype con ex subse o a
me ically con ex me ic space M. Then gi en any ε > 0 he e exis s a
nonexpansi e e ac ion R:M→Awi h he p ope y ha i u∈M A
he e exis s ∈M Awi h d( , R( )) = dis ( , A)and d(u, )≤ε.
Gi en Aand Mas abo e he ε-le el se o Awi h espec o Mis de ined
as ollows:
Sε={ ∈M: dis ( , A) = ε}.
The ollowing co olla y, al hough no s a ed in [3], is howe e a consequence
o he p oo o he p e ious heo em.
Co olla y 3.2. Le ε > 0,Aand Mas abo e, and S=Sn∈NSnε, hen he
e ac ion gi en by Theo em 3.1 can be chosen so ha d( , R( )) = dis ( , A)
o any ∈S.
On selec ions o he me ic p ojec ion... 13
Ou i s esul is he nex echnical lemma on Theo em 3.1.
Lemma 3.3. Le Abe a weakly ex e nally hype con ex subse o a me ically
con ex me ic space M. Fo each n∈Nle εn=1
2n, hen he e exis s a
nonexpansi e e ac ion n(associa ed o εn) as in Co olla y 3.2 such ha
he sequence o e ac ions { n}sa is ies ha
d( n(x), m(x)) ≤
j=m
X
j=n+1
1
2j
o x∈Mand n<m.
P oo . Fo n= 1 we ake 1as he one gi en by Co olla y 3.2. We p o e
nex ha gi en i o 1≤i≤nas in he s a emen o he lemma we can con-
s uc n+1 as equi ed. We conside Sεn+1 and p oceed as in Theo em 3.1.
A e applying Zo n’s Lemma we may assume ha Hεn+1 is he maximal
subse o Sεn+1 whe e n+1 can be ex ended as equi ed. Then we need o
p o e ha Sεn+1 =Hεn+1 . Suppose ha he e exi s ∈Sεn+1 Hεn+1 and
le
P( ) =
x∈A
B(x, d (x, ))!∩
u∈Hεn+1
B( n+1 (u), d (u, ))!
∩B , 1
2n+1 ∩B n( ),1
2n+1 ∩A.
All we need o p o e is ha P( )6=∅. Since Ais weakly hype con ex and
only one o he abo e balls is cen e ed ou side A, i is enough o check ha
each wo o such balls ha e nonemp y in e sec ion. In a case-by-case check
i only es s o s udy hose cases in ol ing he ball cen e ed a n( ), o he
cases we e al eady s udied in [3]. Fo hese cases i is enough o ecall ha
d(x, n( )) ≤d(x, ) o x∈A, now, since Ais p oximinal, le p ∈Asuch
ha d( , p ) = dis ( , A), so
d( , n( )) ≤d( , p ) + d(p , n( ))
≤2 dis ( , A) = 1
2n,
and inally, o u∈Hεn+1 ,
d( n+1(u), n( )) ≤d( n+1(u), n(u))+d( n(u), n( ))
(by induc ion hypo hesis)
≤1
2n+1 +d(u, ).
So we can conside n+1 de ined on he whole Sεn+1 as equi ed. Nex we
show how o ex end n+1 o A∪Sεn+1 ∪S2εn+1 . Le ∈S2εn+1 =Sεn, hen

14 R. Esp´ınola
he se
P( ) =
x∈A
B(x, d (x, ))!∩
u∈Sεn+1
B( n+1 (u), d (u, ))!
∩B , 1
2n∩B n( ),1
2n+1 ∩A
is nonemp y since d( n( ), x)≤d( , x) o x∈A,d( n+1(u), n( )) ≤
d( n+1(u), n(u)) + d( n(u), n( )) ≤(by induc ion) 1
2n+1 +d(u, ), and,
since he me ic con exi y o Mimplies ha he e exis s ˆ ∈Sεn+1 such
ha d( , ˆ ) = 1
2n+1 , we ha e
d( , n( )) ≤d( , ˆ ) + d(ˆ , n(ˆ ))+d( n(ˆ ), n( ))
≤1
2n+1 +1
2n+1 +1
2n+1 =1
2n+1
2n+1 .
Now, by selec ing a poin in P( )i is possible o ex end n+1 as equi ed
om Sεn+1 o Sεn+1 ∪ { }. This same a gumen shows how o ex end n+1
o A∪Sεn+1 ∪S2εn+1 on o Aas equi ed.
Le S=S∞
i=1 Siεn+1 . By p oceeding as abo e bu selec ing ˆ ∈S(i−1)εn+1
o ∈Siεn+1 , and using induc ion i ollows ha he e exis s a nonexpansi e
e ac ion n+1 o A∪Son o Aas equi ed. Le ∈M (A∪S), hen we
conside he se
P( ) =
x∈A∪S
B( n+1(x), d(x, ))!∩B n( ),1
2n+1 .
P( )is nonemp y om he hype con exi y o Aand he ac ha , by in-
duc ion hypo hesis,
d( n+1(x), n( )) ≤d( n+1(x), n(x)) + d( n(x), n( ))
≤d(x, ) + 1
2n+1 .
Again, using induc ion i ollows ha n+1 can de ined on Mas equi ed.
Hence
d( n+1(x), n(x)) ≤1
2n+1
o x∈M. Now le { n}be he sequence o e ac ions gi en by he abo e
p ocedu e, hen o m > n and x∈M
d( m(x), n(x)) ≤
j=m
X
j=n+1
d( j(x), j−1(x))
≤
j=m
X
j=n+1
1
2j.
On selec ions o he me ic p ojec ion... 15
Hence he p oo o he lemma is comple ed. 
The ollowing co olla y is an immedia e consequence o he p e ious
lemma.
Co olla y 3.4. Le { n}be he sequence o e ac ions gi en by Lemma 3.3,
hen { n(x)}is con e gen o each x∈M.
P oo . To p oo his co olla y i is enough o ecall ha hype con ex spaces
a e comple e, hence Ais comple e. 
Nex we p esen he main esul o his wo k.
Theo em 3.5. Le Abe a comple e weakly ex e nally hype con ex subse o
a me ically con ex me ic space M, hen Ais a p oximinal nonexpansi e
e ac o M.
P oo . Le { n}be he sequence o e ac ions gi en by Lemma 3.3, hen
we de ine he mapping :M→Aas
(x) = lim
n→∞ n(x).
Co olla y 3.4 implies ha is a well-de ined e ac ion on A. Mo eo e ,
since nis nonexpansi e o each n∈N, is nonexpansi e. Addi ionally we
claim ha d( (x), x) = dis (x, A) o x∈M. Fo x∈A he e is no hing o
p o e, so le x∈M A. Fo each n∈N he e exis s n∈M Asuch ha
d(x, n)≤1
2nand
d( n, n( n)) = dis ( , A).
Hence we ha e
d(x, n(x)) ≤d(x, n) + d( n, n( n))+d( n( n), n(x))
≤1
2n+ dis ( n, A) + 1
2n
≤1
2n+ dis (x, A) + d( n, x) + 1
2n
=3
2n+ dis (x, A).
Taking limi as n→ ∞ he conclusion ollows. 
Since Theo em 2.1 o [3] implies ha p oximinal nonexpansi e e ac s
o hype con ex spaces a e weakly ex e nally hype con ex, he p e ious he-
o em can be e-w i en in he ollowing way.
Theo em 3.6. Le Mbe a hype con ex me ic space and le A⊆Mbe
nonemp y. Then Ais a p oximinal nonexpansi e e ac o Mi , and only
i , Ais a weakly ex e nally hype con ex subse o M.
16 R. Esp´ınola
Nex we gi e applica ions o Theo ems 3.5–3.6. We begin wi h an appli-
ca ion o he exis ence o bes p oximi y pai s, in pa icula we ha e he
ollowing ex ension o P oposi ion 2.8 in [5].
Co olla y 3.7. Le Mbe a hype con ex me ic space and le Aand Bbe
nonemp y weakly ex e nally hype con ex subse s o M. Then A0and B0a e
nonemp y and hype con ex.
P oo . The same p oo o P oposi ion 2.8 in [5] ca ies o e since Aand B
a e p oximinal nonexpansi e e ac s and, om Lemma 2.8, Nε(A)is weakly
ex e nally hype con ex and he e is and ε-cons an nonexpansi e e ac ion
o Nε(A)on A.
This co olla y allows us o answe in he posi i e a ques ion aised in [5],
mo e p ecisely we ob ain he ollowing ex ension o Theo em 2.10 in [5].
Theo em 3.8. Le Aand Bbe wo weakly ex e nally hype con ex subse s
o a hype con ex me ic space Mwi h Abounded, and suppose T∗:A→2B
is such ha :
(i) o each x∈A,T∗(x)is a nonemp y admissible (mo e gene ally,
ex e nally hype con ex) subse o B;
(ii) T∗: (A, d)→(2B, dH)is nonexpansi e (whe e dHis he Hausdo
me ic);
(iii) T∗(A0)⊆B0.
Then he e exis s x0∈Asuch ha
dis (x0, T∗(x0)) = dis (A, B) = in {dis (x, T∗(x)) : x∈A}.
P oo . The p oo o his heo em ollows he same s eps as ha o Theo-
em 2.10 in [5]. 
We inish his wo k wi h he nonexpansi e e sion o he Fan’s app ox-
ima ion p inciple gi en in [3] (Theo em 5.4). We omi i s p oo since i
ollows in a simila way as he p oo o Theo em 5.4 in [3].
Co olla y 3.9. Le Abe a bounded weakly ex e nally hype con ex subse o
a hype con ex me ic space Mand suppose ha T:A→Mis a nonexpan-
si e mapping. Then he e exis s x∈Asuch ha
d(x, T(x)) = in {d(y, T (x)) : y∈A}.
Re e ences
[1] A onszajn, N., P. Pani chpakdi, Ex ensions o uni o mly con inuous ans o ma ions
and hype con ex me ic spaces, Paci ic J. Ma h. 6(1956), 405–439.
[2] Esp´ınola, R., M. A. Khamsi, In oduc ion o hype con ex spaces, Handbook o Me -
ic Fixed Poin Theo y, (W. A. Ki k, B. Sims, eds.), Kluwe Academic Publishe s,
Do d ech –Bos on–London, 2001, pp. 391–435.
[3] Esp´ınola, R., W. A. Ki k and G. López, Nonexpansi e e ac s in hype con ex spaces,
J. Ma h. Anal. Appl. 251 (2000), 557–570.
On selec ions o he me ic p ojec ion... 17
[4] Khamsi, M. A., W. A. Ki k and C. Ma ´ınez Y´a˜nez, Fixed poin and selec ion heo ems
in hype con ex spaces, P oc. Ame . Ma h. Soc. 128 (2000), 3275–3283.
[5] Ki k, W. A., S. Reich and P. Vee amani, P oximinal e ac s and bes p oximi y pai
heo ems, Num. Func . Anal. Op . 24 (2003), 851–862.
[6] Sine, R., Hype con exi y and app oxima e ixed poin s, Nonlinea Anal. 13 (1989),
863–869.
Ra a Esp´ınola
Depa amen o de An´alisis Ma em´a ico
Uni e sidad de Se illa
P.O. Box 1160
Se ille, Spain
e-mail: [email p o ec ed]
Recei ed Janua y 25, 2005