INTRODUCTION TO HYPERCONVEX SPACES
R. ESP´
INOLA AND M. A. KHAMSI
Con en s
1. P e ace 1
2. In oduc ion and basic de ini ions 2
3. Some basic p ope ies o Hype con ex spaces 4
4. Hype con exi y, Injec i i y and Re ac ion 8
5. Mo e on Hype con ex spaces 15
6. Fixed poin p ope y and Hype con exi y 20
7. Topological ixed poin heo ems and Hype con exi y 23
8. Isbell’s Hype con ex Hull 27
9. Se - alued mappings in Hype con ex spaces 30
10. The KKM heo y in Hype con ex spaces 35
11. Lambda-Hype con exi y 38
Re e ences 40
1. P e ace
The no ion o hype con exi y is due o A onszajn and Pani chpakdi [1] (1956) who p o ed
ha a hype con ex space is a nonexpansi e absolu e e ac , i.e. i is a nonexpansi e e ac
o any me ic space in which i is isome ically embedded. The co esponding linea heo y is
well de eloped and associa ed wi h he names o Gleason, Goodne , Kelley and Nachbin (see o
ins ance [19, 29, 42, 46]). The nonlinea heo y is s ill de eloping. The ecen in e es in o hese
spaces goes back o he esul s o Sine [54] and Soa di [57] who p o ed independen ly ha ixed
poin p ope y o nonexpansi e mappings holds in bounded hype con ex spaces. Since hen
many in e es ing esul s ha e been shown o hold in hype con ex spaces.
Recall also ha Jawha i, Misane and Pouze [27] we e able o show ha Sine and Soa di’s
ixed poin heo em is equi alen o he classical Ta ski’s ixed poin heo em in comple e o de ed
se s. This happens ia he no ion o gene alized me ic spaces. The e o e, he no ion o hype -
con exi y should be unde s ood and app ecia ed in a mo e abs ac o mula ion. I is no ou
pu pose, howe e , o s udy hype con ex spaces om his mo e gene al poin o iew, in e es ed
eade s may consul he e e ences [22, 27, 28, 38, 59].
Along his chap e we will desc ibe and s udy some o he mos cha ac e is ic p ope ies ha
hype con ex spaces enjoy. In opposi ion o he lack o linea i y hype con exi y p o ides us wi h
a eally ich me ic s uc u e ha leads o a collec ion o su p ising and beau i ul esul s ela ed
o di e en b anches o ma hema ics as, o ins ance, opology, g aph heo y, mul i alued anal-
ysis, ixed poin heo y,... I is ou aim o p esen some o hese esul s abou hype con exi y
emphasizing hei ela ion o ixed poin heo y.
1
2 R. ESP´
INOLA AND M. A. KHAMSI
This chap e has been di ided in o ele en sec ions. We begin wi h basic de ini ions and p op-
e ies which will lead us o a i s app oach o he ela ion be ween hype con exi y and he
Hahn-Banach heo em. In Sec ion 3 we con inue wi h mo e gene al p ope ies on hype con-
exi y, we will lea n some undamen al ac s o he geome y o hype con ex spaces, ac s such
as ha e e y hype con ex space is comple e o ele an p ope ies e i ied by hei Chebyshe
elemen s. In his sec ion we will also dis inguish among ou di e en impo an subclasses o
hype con ex subse s o a hype con ex space, namely hype con ex, admissible, ex e nally and
weakly ex e nally hype con ex subse s. These classes o se s will be o g ea impo ance all along
ou ea men and mo e pa icula ly when ixed poin esul s will be s a ed. In Sec ion 4 we
ela e hype con exi y o injec i i y and absolu e e ac s. We will see he ole ha hese la e
concep s played in he mo i a ions o hype con exi y o inish wi h mo e ecen and sub le esul s
on he exis ence o e ac ions and ε-cons an nonexpansi e e ac ions. Sec ion 5 will be de o ed
o he s udy o deepe ac s o he geome y o hype con ex spaces, we begin by s udying he bad
p ope ies o hese spaces wi h espec o he in e sec ion, s a ing ha hype con ex subse s o a
hype con ex spaces do no de ine a closed (unde he in e sec ion o se s) class o se s. This is
in pa co ec ed by a e y celeb a ed esul due o Baillon in [3] on in e sec ion o hype con ex
se s ha allows o de ine he concep o “hype con ex hull” as ha one o “injec i e hull” gi en
by Isbell in [24]. This sec ion is inished wi h he de ini ion o he noncompac ness measu es o
Hausdo and Ku a owski as well as hei p ope ies in hype con ex spaces.
We abandon gene ali ies on hype con exi y in Sec ion 6 o s udy some mo e ecen and pa -
icula p ope ies ha will lead us o deep ixed poin esul s. The exis ence o ixed poin o
nonexpansi e mappings is s udied as well as simila esul s on amily o commu ing mappings.
This sec ion is closed by a non-elsewhe e published esul on asymp o ically nonexpansi e map-
pings. Sec ion 7 deals wi h ano he kind o ixed poin heo ems, hose in which compac ness
condi ions a e conside ed. We s udy how di e en esul s s a ed in linea spaces a e s ill ue in a
hype con ex se ing. Sec ion 8 is comple ely de o ed o he s udy o he “injec i e hull” o Isbell
and ex emal unc ions, his concep is one o he mos in e es ing and in iguing ones in hype -
con ex me ic spaces and well dese es a whole sec ion o i s be e unde s anding. Ano he
e y impo an cha ac e is ic o hype con ex spaces is s udied in Sec ion 9, in his sec ion we
ocus on mul i alued mappings and s a e a su p ising esul on selec ion o mul i alued mappings
ha implies di e en esul s on ixed poin heo y and exis ence o nonexpansi e selec ions o he
me ic p ojec ion. The s udy on mul i alued mappings will be comple ed in Sec ion 10 whe e he
KKM p inciple is adap ed o hype con ex spaces and new e sions o classical esul s on ixed
poin heo y a e ob ained. The concep o “lambda hype con exi y” is s udied in he las sec ion
o he chap e , Sec ion 11. This is a ecen ly in oduced [34] idea which may be unde s ood as
an ex ension o he geome ical de ini ion o hype con exi y inspi ed in o me s udies due o
G nbaum [20].
Finally we wan o poin ou ha he ma e ial we p esen he e goes om classical o e y ecen
ac s ha will lead he eade o an upda ed knowledge abou ixed poin esul s on hype con ex
spaces. The eade will ind, howe e , a la ge collec ion o i ems a he end o he chap e om
which i is possible o con inue he s udy o ac s and ela ed subjec s ha we e no ea ed in
de ail in his chap e .
2. In oduc ion and basic de ini ions
No doub ha Hahn-Banach heo em played a majo ole in unc ional analysis. In ac , i
is qui e impossible o hink o Banach spaces wi hou his heo em. So i was clea om he
beginning ha an ex ension o his heo em o me ic spaces was o be ound. The i s o s udy
INTRODUCTION TO HYPERCONVEX SPACES 3
his ques ion we e A onszajn and Pani chpakdi in [1]. Thei in es iga ion led o he disco e y
o hype con ex me ic spaces. In o de o app ecia e hei indings, one needs o emembe he
p oo o Hahn-Banach heo em.
Theo em 2.1. Le Xbe a eal ec o space, Ybe a linea subspace o X, and ρa semino m on
X. Le be a linea unc ional de ined on Ysuch ha (y)≤ρ(y), o all y∈Y. Then he e
exis s a linea unc ional gde ined on X, which is an ex ension o (i.e. g(y) = (y), o all
y∈Y), which sa is ies g(x)≤ρ(x), o all x∈X.
P oo . The main a gumen behind he p oo o his heo em is he s uc u e o he eal line
R. Indeed, ia Hausdo maximali y p inciple, i is enough o ex end o Y+R·x0, whe e
x0∈X−Y. So we need o ind g(x0) such ha
g(y+αx0) = g(y) + αg(x0) = (y) + αg(x0)
g(y+αx0)≤ρ(y+αx0)
o any eal numbe α∈Rand any y∈Y. Since ρis a semino m and is a linea unc ional,
we may assume α=±1. This means ha wha we need o ind is a numbe A(which e en ually
will be equal o g(x0)), such ha (y)±A≤ρ(y±x0),which ansla es o (y)−ρ(y−x0)≤
A≤ρ(y∗+x0)− (y∗) o any y, y∗∈Y. In o he wo ds, we mus ha e
y,y∗∈Yh (y)−ρ(y−x0), ρ(y∗+x0)− (y∗)i6=∅.
Se Iy,y∗=h (y)−ρ(y−x0), ρ(y∗+x0)− (y∗)i, o y, y∗∈Y. Then using he linea i y o
and he semino m beha io o ρ, i is easy o check ha o any y1,y2,y∗
1, and y∗
2in Y,
Iy1,y∗
1∩Iy1,y∗
16=∅.The p oo will be comple e i we use he ollowing well-known undamen al
p ope y o he eal line R:
“I {Iα}α∈Γis a collec ion o in e als such ha Iα∩Iβ6=∅, o any α, β ∈Γ, hen we ha e
α∈Γ
Iα6=∅”.
I is his p ope y ha is a he hea o he new concep disco e ed by A onszajn and Pan-
i chpakdi. No e ha an in e al may also be seen on he eal line as a closed ball. Indeed,
he in e al [a, b] is also he closed ball cen e ed a (a+b)/2 wi h adius = (b−a)/2, i.e.
[a, b] = B(a+b
2,b−a
2).
So he abo e in e sec ion p ope y may also be seen as a ball in e sec ion p ope y. This is
qui e in e es ing since in me ic spaces i is na u al o alk abou balls. Bu keep in mind ha
in o de ed se s o example, in e als a e mo e na u al han balls.
Th oughou his chap e , he balls e e ed o a e closed. The e o e we will omi he wo d
closed.
Rema k 2.2. Le Mbe a me ic space. Using he iangle inequali y, we ha e B(x1, 1)∩
B(x2, 2)6=∅implies d(x1, x2)≤ 1+ 2 o any x1, x2∈Mand posi i e numbe s 1, 2. The
con e se is ue on he eal line and co esponds o he Menge con exi y in me ic spaces.
De ini ion 2.3. Le Mbe a me ic space. We say ha Mis me ically con ex i o any poin s
x, y ∈Mand posi i e numbe s αand βsuch ha d(x, y)≤α+β, he e exis s z∈Msuch ha
d(x, z)≤αand d(z, y)≤β, o equi alen ly z∈B(x, α)∩B(y, β).
4 R. ESP´
INOLA AND M. A. KHAMSI
The e o e, Mis me ically con ex i B(x, α)∩B(y, β)6=∅i and only i d(x, y)≤α+β o
any poin s x, y ∈Mand posi i e numbe s αand β.
Rema k 2.4. No e ha some au ho s de ine me ic con exi y sligh ly di e en ly. Indeed, (M, d)
is me ically con ex i and only i o any poin s x, y ∈Mand any numbe α∈[0,1], he e exis s
z∈Msuch ha d(x, z) = αd(x, y)and d(y, z) = (1 −α)d(x, y).This de ini ion is ha d o
ex end o gene al s uc u es since i uses he mul iplica ion ope a ion, which is, o example, ha d
o de ine in disc e e se s.
The abo e discussion shows ha he Hahn-Banach ex ension heo em is closely ela ed o
an in e sec ion p ope y o he closed balls combined wi h some kind o me ic con exi y. The
concep o hype con exi y in oduced by A onszajn and Pani chpakdi cap u ed hese ideas.
De ini ion 2.5. The me ic space Mis said o be hype con ex i
α∈Γ
B(xα, α)6=∅ o any
collec ion o poin s {xα}α∈Γin Mand posi i e numbe s { α}α∈Γsuch ha d(xα, xβ)≤ α+ β
o any αand βin Γ.
3. Some basic p ope ies o Hype con ex spaces
Clea ly om he p e ious sec ion, he eal line Ris hype con ex. In ac , we can easily p o e
ha he in ini e dimensional Banach space l∞is hype con ex. One way o see ha is o use he
ollowing esul .
Theo em 3.1. Le (Mα, dα)α∈Γbe a collec ion o hype con ex me ic spaces. Conside he
p oduc space M=Q
α∈Γ
Mα. Fix a= (aα)∈ M and conside he subse Mo Mde ined by
M=(xα)∈ M; sup
α∈Γ
dα(xα, aα)<∞.
Then (M, d∞)is a hype con ex me ic space whe e d∞is de ined by
d∞(xα),(yα)= sup
α∈Γ
dα(xα, yα)
o any (xα),(yα)∈M.
This heo em is a s uc u al esul . I s p oo is an easy consequence om he ac ha o any
ball B((xα), ) in M, we ha e B((xα), ) = Y
α∈Γ
Bα(xα, ) whe e Bα(xα, ) is he ball cen e ed a
xαwi h adius in Mα.
We will see la e on ha any me ic space may be embedded in a “small” hype con ex me ic
space. Tha cons uc ion, disco e ed by Isbell, is no immedia e and ha d o g asp. Bu o
many cases, i is wo h o know ha any me ic space may be embedded isome ically in o a
hype con ex me ic space. This is easy o see. Indeed, le (M, d) be a me ic space. Se
l∞(M) = {(xm)m∈M∈RM; sup
m∈M|xm|<∞}.
On l∞(M) de ine he dis ance d∞by d∞((xm),(ym)) = sup
m∈M|xm−ym|.The me ic space
(l∞(M), d∞) is hype con ex. To see ha Membeds isome ically in o l∞(M), ix a∈Mand
conside he map I:M→l∞(M) de ined by I(b) = d(b, m)−d(a, m)m∈M o any b∈M. I
INTRODUCTION TO HYPERCONVEX SPACES 5
is easy o check ha d∞I(b), I(c)=d(b, c) o any b, c ∈M.
Nex we discuss comple eness o hype con ex me ic spaces. In ac , a weake e sion o he
bina y-ball in e sec ion p ope y is needed o insu e he comple eness o he me ic space. In-
deed, we will say ha he me ic space Mhas he ball in e sec ion p ope y (BIP in sho ) i
α∈Γ
Bα6=∅ o any collec ion o balls (Bα)α∈Γsuch ha T
α∈Γ
Bα6=∅, o any ini e subse Γ ⊂Γ.
P oposi ion 3.2. Any me ic space Mwhich has he ball in e sec ion p ope y is comple e. In
pa icula any hype con ex me ic space is comple e.
P oo . Le (xn) be a Cauchy sequence in M. Fo any n≥1, se n= sup
m≥n
d(xn, xm). Conside
he collec ion o balls B(xn, n)n≥1. Since o m≥nwe ha e d(xn, xm)≤ n, hen
xnk∈B(xn1, n1)∩B(xn2, n2)∩···∩B(xnk, nk)
o any n1< n2<··· < nk, bu Mhas he ball in e sec ion p ope y so we may conclude ha
T
n≥1
B(xn, n)6=∅. Now, since (xn) is a Cauchy sequence, lim
n→∞ n= 0 and so he in e sec ion
T
n≥1
B(xn, n) is educed o one poin zwhich is he limi o he sequence (xn).
We ha e jus seen a p ope y abou in e sec ion o balls a he han abou hype con exi y.
Ball in e sec ion p ope ies ha e been ex ensi ely s udied in connec ion wi h di e en p oblems
as geome ical p ope ies o Banach spaces o ex ension o mappings. In e es ed eade s may ind
mo e on his in e es ing subjec in [6, 23, 43, 45].
A his poin we in oduce some no a ion which will be used h oughou he emainde o his
wo k. Fo a subse Ao a me ic space M, se :
x(A) = sup{d(x, y) : y∈A}, x ∈M;
(A) = in { x(A) : x∈M};
R(A) = in { x(A) : x∈A};
diam(A) = sup{d(x, y) : x, y ∈A};
C(A) = {x∈M: x(A) = (A)};
CA(A) = {x∈A: x(A) = (A)};
co (A) = T{B:Bis a ball and B⊇A}.
(A) is called he adius o A( ela i e o M), diam(A) is called he diame e o A, R(A) is
called he Chebyshe adius o A, C(A) is called he cen e o A(in M), CA(A) is called he
Chebyshe cen e o A, and co (A) is called he co e o A.
We now p o e a echnical lemma.
Lemma 3.3. Suppose Ais a bounded subse o a hype con ex me ic space M. Then:
1. co (A) = T{B(x, x(A)) : x∈M}.
2. x(co (A)) = x(A), o any x∈M.
3. (co (A)) = (A).
4. (A) = 1
2diam(A).
5. diam(co (A)) = diam(A).
6. I A=co (A), hen (A) = R(A). In pa icula we ha e R(A) = 1
2diam(A).

6 R. ESP´
INOLA AND M. A. KHAMSI
P oo . 1. Since B(x, x(A)) con ains A o each x∈Mi mus be he case ha
co (A)⊆ nB(x, x(A)) : x∈Mo.
On he o he hand, i A⊆B(x, ) hen x(A)≤ so B(x, x(A)) ⊆B(x, ). Hence
{B(z, z(A)) : z∈M} ⊆ B(x, ).
This clea ly implies
co (A) = {B(x, x(A)) : x∈M}.
2. By 1, x(co (A)) = sup nd(x, y) : y∈T
z∈M
B(z, z(A))oso, in pa icula , y∈co (A) implies
y∈B(x, x(A)) o any x∈M. Hence d(x, y)≤ x(A), which p o es x(co (A)) ≤ x(A).The
e e se inequali y is ob ious since A⊆co (A).
3. This is immedia e om he de ini ion o .
4. Le δ= diam(A) and conside he amily nBa, δ
2:a∈Ao.I a, b ∈A hen d(a, b)≤
δ=δ
2+δ
2so by hype con exi y
a∈A
Ba, δ
26=∅.
I xis any poin in his in e sec ion hen d(x, a)≤δ
2so x(A)≤δ
2.On he o he hand d(a, b)≤
d(a, z) + d(z, b) o any a, b ∈Aand z∈Mso δ≤2 z(A) om which δ≤2 (A).The e o e
δ≤2 (A)≤2 x(A)≤δp o ing (A) = δ
2.
5. Using 3 and 4, diam(A) = 2 (A) = 2 co (A)= diam(co (A)).
6. No e ha we always ha e 1
2diam(A)≤ (A)≤R(A).We may w i e A=T
i∈I
Biwhe e Biis
a closed ball o any i∈I. Now, since
a∈A
Ba, δ
26=∅whe e δ= diam(A), i is easy o check
ha any wo balls d awn om he collec ion
nBi, i ∈Io[nBa, δ
2;a∈Ao
ha e nonemp y in e sec ion. Recalling now he hype con exi y o M,
C=A nBa, δ
2;a∈Ao= nBi;i∈Io nBa, δ
2;a∈Ao6=∅.
Le x∈ C. Then x(A)≤δ/2 and he e o e δ
2≤ (A)≤R(A)≤ x(A)≤δ
2,which clea ly
implies (A) = R(A) = 1
2diam(A).
De ini ion 3.4. Le Mbe a me ic space. By A(M)we deno e he collec ion o all subse s o
Mwhich a e in e sec ion o balls, i.e. A(M) = {A⊂M;A=co (A)}.The elemen s o A(M)
a e called admissible subse s o M.
INTRODUCTION TO HYPERCONVEX SPACES 7
I is clea ha A(M) con ains all he closed balls o Mand is s able by in e sec ion, i.e. he
in e sec ion o any collec ion o elemen s om A(M) is also in A(M). F om he abo e esul s,
o any A∈ A(M), we ha e
C(A) =
a∈A
Ba, R(A) A∈ A(M).
Mo eo e , diam(C(A)) ≤diam(A)/2. So we ha e A=C(A) i and only i A∈ A(M) and
diam(A) = 0, i.e. Ais educed o one poin .
This p ope y and he abo e s udied ones a e ex emely impo an when we discuss he ixed
poin p ope y in hype con ex me ic spaces. In ac , we will show in his chap e ha admissible
subse s in hype con ex me ic spaces enjoy some nice p ope ies. Bu admissible subse s will no
be he only class o subse s ha will be o in e es o us, le us in oduce h ee mo e classes o
subse s ha will be o g ea impo ance in ou exposi ion.
De ini ion 3.5. 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, whe e dis (x, E) = in {d(x, y) : y∈E}, hen i ollows
ha T
α
B(xα, α)∩E6=∅.The class o all he ex e nally hype con ex subse s o Mwill be deno ed
as E(M).
De ini ion 3.6. 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. Mo e p ecisely,
gi en any amily {xα}o poin s in Mall 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 ∩αB(xα; α)∩E6=∅.The class o all he weakly ex e nally hype con ex subse s o Mwill
be deno ed as W(M).
Addi ionally, we will deno e he class o hype con ex subse s o a me ic space Mas H(M).
We inish his sec ion s udying he ela ion among hese classes o se s. In o de o do ha
we need o in oduce he concep o p oximali y.
De ini ion 3.7. A subse Eo a me ic space Mis said o be p oximinal (wi h espec o M) i
he in e sec ion E∩B(x, dis (x, E)) is nonemp y o each x∈M.
Lemma 3.8. I Eis ei he an admissible, ex e nally hype con ex o weakly ex e nally hype con-
ex subse o a hype con ex me ic space M, hen Eis p oximinal in M.
P oo . We w i e he p oo o he case E=Aan admissible subse . O he cases a e simila . Se
A=T
i∈I
Bi. Then o any ε > 0, he e exis s aε∈Asuch ha d(x, aε)≤dis (x, A) + ε. Clea ly
his implies Ti∈IBiTB(x, dis (x, A) + ε)6=∅.Since Mis hype con ex, hen we mus ha e
A∩B(x, dis (x, A)) =
i∈I
Bi 
ε>0
B(x, dis (x, A) + ε)6=∅
which comple es he p oo ha Ais p oximinal.
Rema k 3.9. No ice ha we only need M o be hype con ex in he p o ed case, i.e. he same
esul o ex e nally o weak ex e nally hype con ex subse s does no equi e hype con exi y o he
8 R. ESP´
INOLA AND M. A. KHAMSI
space M. This is essen ially due o he ac ha o hese o he subse s he hype con ex hypo hesis
is g an ed by he de ini ion o hese se s.
Theo em 3.10. Le Mbe a hype con ex me ic space, hen A(M)⊆ E(M)⊆ W(M)⊆ H(M).
P oo . A(M)⊆ E(M): Le Abe an admissible subse o Mand le {xα}α∈Γbe a amily o poin s
in Mand { α}α∈Γbe amily o eal numbe s sa is ying d(xα, xβ)≤ α+ βand dis (xα, A)≤ α
o any α, β ∈Γ. Since Ais p oximinal, o any α∈Γ, he e exis s aα∈Asuch ha d(xα, aα) =
dis (xα, A), which gi es A∩B(xα, α)6=∅. Since Mis hype con ex, he condi ions on bo h
amilies imply ∩α∈ΓB(xα, α)6=∅. Since Ais admissible and A∩B(xα, α)6=∅i ollows
A 
α∈Γ
B(xα, α)6=∅
which p o es he i s inclusion.
E(M)⊆ W(M): I ollows di ec ly om de ini ion.
W(M)⊆ H(M): I ollows di ec ly om de ini ion.
Rema k 3.11. No ice ha i in he de ini ion o weak ex e nally hype con exi y we impose he
condi ion ha one o he balls mus be ou o E, hen weakly ex e nally hype con ex subse s do
no ha e why o be hype con ex. Think o R2minus he in e io o he uni squa e wi h he sup e-
mum no m and ake Eas he bo de o he uni squa e. Ewould be weakly ex e nally hype con ex
unde his new de ini ion bu no hype con ex.
Rema k 3.12. Addi ionally, he amilies A(M),E(M),W(M), and H(M)do no coincide in
gene al. Le Mbe he hal igh eal plane endowed wi h he maximum me ic, i.e. M={(x, y)∈
R2:x≥0}wi h he maximum me ic. Le E=B((0,−1),2) ∩M,W={(x, x) : 0 ≤x≤1},
and H={(x, x
2) : 0 ≤x≤1} hen Eis ex e nally hype con ex in Mbu no admissible, Wis
weakly ex e nally hype con ex in Mbu no ex e nally hype con ex, and, inally, His hype con ex
bu no weakly ex e nally hype con ex.
4. Hype con exi y, Injec i i y and Re ac ion
In his sec ion, we will discuss A onszajn and Pani chpakdi ideas on how hype con exi y
cap u es Hahn-Banach ex ension heo em in me ic spaces. Be o e we s a e A onszajn and Pan-
i chpakdi’s main esul , we ecall he de ini ion o nonexpansi e mappings.
De ini ion 4.1. Le (M1, d1)and (M2, d2)be me ic spaces. A map T:M1→M2is said o be
Lipschi zian i he e exis s a cons an k≥0such ha
d2(T(x), T(y)) ≤kd1(x, y)
o any x, y ∈M1. I k= 1, he map is called nonexpansi e (and con ac ion i k < 1).
A me ic space Mis said o be injec i e i i has he ollowing ex ension p ope y: Whene e
Yis a subspace o Xand :Y→Mis nonexpansi e, hen has a nonexpansi e ex ension
˜
:X→M. This ac has se e al nice consequences.
Theo em 4.2. Le Hbe a me ic space. The ollowing s a emen s a e equi alen :
(i) His hype con ex;
INTRODUCTION TO HYPERCONVEX SPACES 9
(ii) His injec i e.
P oo . Fi s assume His hype con ex. Le Dbe a me ic space and T:D→ Hbe nonexpansi e.
Le Mbe a me ic space con aining Dme ically. Conside he ollowing se
C={(TF, F); TF:F→Hwi h D⊂F⊂Mme ically}
whe e TFis a nonexpansi e ex ension o T. We ha e (T, D)∈ C. The e o e, Cis no emp y. On
he o he hand, one can o de pa ially Cby (TF, F )≺(TG, G) i and only i F⊂Gand he
es ic ion o TG o Fis TF.
I is easy o see ha Csa is ies he hypo hesis o Zo n’s lemma. The e o e, Chas maximal
elemen s. Le (T1, F1) be one maximal elemen o C. Le us show ha F1=M. Assume no .
Le z∈M F1and se F=F1∪{z}. Le us ex end T1 o F. The ques ion is o ind a poin
z1, which will play he ole o he alue o he ex ension a z. Since we need he ex ension o be
nonexpansi e, we mus ha e dT1(x), z1≤d(x, z) o all x∈F1. Conside he amily o closed
balls nBT1(x), d(x, z)o, wi h x∈F1. Since dT1(x), T1(y)≤d(x, y)≤d(x, z) + d(z, y) o
all x, y ∈F1, he hype con exi y o Himplies ∩x∈F1BT1(x), d(x, z)6=∅.Le z1be any poin
in his in e sec ion. Se T∗:F→Hby
T∗(x) = T1(x) i x6=z
z1i x=z.
I is easy o check ha (T∗, F) belongs o C, hence (T1, F1)≺(T∗, F ) and (T1, F1)6= (T∗, F).
This con adic s he maximali y o (T1, F1). The e o e, F1=M. In o he wo ds, Thas a nonex-
pansi e ex ension o M.
Con e sely, assume ha His injec i e, i.e. o e e y me ic space Dand e e y nonexpansi e
map T:D→H, he e exis s a nonexpansi e ex ension T∗:M→Ho T, whe e Mis any
me ic space which con ains Dme ically. Le us p o e ha His hype con ex. A onszajn and
Pani chpakdi’s o iginal p oo is di ided in o wo pa s. Fi s , hey showed ha His me ically
con ex. Then hey showed ha i is hype con ex. He e, we will y o a emp a p oo using
Isbell’s ideas. Indeed, gi en {xα}α∈Γin Hand posi i e numbe s { α}α∈Γsuch ha d(xα, xβ)≤
α+ β o any αand βin Γ we wan o show ha Tα∈ΓB(xα, α)6=∅.Wi hou any loss o
gene ali y, we may assume ha xα6=xβ o any α6=β. Conside he se Fo posi i e eal
alued unc ions de ined on he se D={xα;α∈Γ}such ha d(xα, xβ)≤ (xα) + (xβ)
o any α, β ∈Γ. No e ha he unc ion :D→Rde ined by (xα) = αbelongs o his se .
Fis pa ially o de ed by he poin wise o de on he eal line. Ob iously, any descending chain
o elemen s o Fhas a lowe bound. Hence, Zo n’s lemma implies he exis ence o a minimal
elemen ∈ F smalle han , i.e. (xα)≤ (xα), o any α∈Γ.
Now, using he minimali y o , we can p o e ha (xα)≤d(xα, xβ) + (xβ) o any αand
βin Γ. Indeed, assume his is no he case. Then he e exis s α0and β0such ha d(xα0, xβ0) +
(xβ0)< (xα0).Se
F(xγ) =  (xγ) i γ6=α0
d(xα0, xβ0) + (xβ0) i γ=α0.
Fsa is ies ha F≤ and F6= , con adic ing he minimali y o . Le ωbe a poin no
in he se H. Conside he se D∗=D∪ {ω}. The dis ance be ween he elemen s o Dis
he one inhe i ed om H. Fo he new poin , se d(ω, xα) = (xα).I is easy o check ha
D∗is a me ic space which con ains Dme ically. Ou assump ion assu es us o he exis ence
o a nonexpansi e ex ension Ro he iden i y map (de ined om Din o H). I is clea ha
16 R. ESP´
INOLA AND M. A. KHAMSI
hen co β(Aβ)∩Aα∈ A(Hα) since Hβ⊂Hα. Hence A0
α∈ A(Hα). The e o e, we ha e A0∈ F.
Since Ais minimal, A=A0which implies
Aα= co β(Aβ)∩Aα, o e e y α≤β .
Le x∈Hβand α≤β. Since Aβ⊂Aα, hen x(Aβ)≤ x(Aα). Because co β(Aβ) =
x∈Hβ
B(x, x(Aβ)), hen we ha e co β(Aβ)⊂B(x, x(Aβ)) which implies x(co β(Aβ)) ≤ x(Aβ).
Addi ionally Aα⊂co β(Aβ) so x(Aβ)≤ x(Aα)≤ x(co β(Aβ)) ≤ x(Aβ). The e o e, we
ha e x(Aα) = x(Aβ) o e e y x∈Hβ. Using he de ini ion o , we ge (Aα)≤ (Aβ).
Le a∈Aαand se s= a(Aα). Then a∈co β(Aβ) since Aα⊂co β(Aβ). Hence a∈
T
x∈Aβ
B(x, s)Tco β(Aβ). So, om he hype con exi y o Hβ,
Sβ=Hβ
x∈Aβ
B(x, s) co β(Aβ)6=∅.
Le z∈Sβ, hen z∈T
x∈Aβ
B(x, s) and, since Aβ=Hβ co β(Aβ), i ollows ha z(Aβ)≤s,
which implies (Aβ)≤s= a(Aα) o e e y a∈Aα. Hence (Aβ)≤ (Aα). The e o e we ha e
(Aβ) = (Aα), o e e y α, β ∈Γ.
Assume ha δ(Aβ)>0 o e e y β∈Γ. Se A00
β=C(Aβ) o e e y β∈Γ. The amily (A00
β)
is dec easing. Indeed, le α≤βand x∈A00
β. Then we ha e x(Aβ) = (Aβ). Since we p o ed
ha z(Aβ) = z(Aα) o e e y z∈Hβ, hen x(Aα) = x(Aβ) = (Aβ) = (Aα),which implies
ha x∈A00
α. The e o e, we ha e A00 =Q
β∈Γ
A00
β∈ F. Since A00 ⊂Aand Ais minimal, we ge
A=A00. The e o e, we ha e C(Aβ) = Aβ o e e y β∈Γ. This con adic s he ac ha Hβis
hype con ex o e e y β∈Γ. Hence he e exis s β0∈Γ such ha δ(Aβ) = 0, o e e y β≥β0.
The p oo o ou claim is he e o e comple e since we ha e Aβ={a} o e e y β≥β0which
clea ly implies ha a∈Tβ∈ΓHβ6=∅.
In o de o comple e he p oo , we need o show ha S=T
β∈Γ
Hβis hype con ex. Le (Bi)i∈I
be a amily o balls cen e ed in Ssuch ha T
i∈I
Bi6=∅. Se Dβ=T
i∈I
BiTHβ o β∈Γ. Since Hβ
is hype con ex and he amily (Bi) is cen e ed in Hβ, hen Dβis no emp y and Dβ∈ A(Hβ).
The e o e, Dβis hype con ex. The abo e p oo shows ha Tβ∈ΓDβ6=∅which comple es he
p oo o Theo em 5.1.
Rema k 5.2. This p oo is di e en om Baillon’s o iginal one. I is li le mo e complica ed.
Bu i has he ad an age o being easy o adap o 1-local e ac . In o he wo ds, he conclusion
o Baillon’s esul holds o 1-local e ac se s.
While he in e sec ion o wo admissible subse s o a gi en hype con ex space is again ad-
missible, in gene al i is no he case ha he in e sec ion o wo hype con ex subspaces o a
hype con ex space is i sel hype con ex, e en i one o hem is admissible. Howe e he ollowing
is ue.
Lemma 5.3. Le Hbe a hype con ex me ic space. Suppose E⊂His ex e nally hype con ex
ela i e o Hand suppose Ais an admissible subse o H. Then E∩Ais ex e nally hype con ex
ela i e o H.
P oo . Suppose {xα}and { α}sa is y d(xα, xβ)≤ α+ βand dis (xα, E ∩A)≤ α.Since Ais
admissible, A=T
i∈I
B(xi; i) and since dis (xα, E ∩A)6=∅i ollows ha d(xα, xi)≤ α+ i o

INTRODUCTION TO HYPERCONVEX SPACES 17
each i∈I. Also, since A⊂B(xi; i),i ollows ha dis (xi, E∩A)≤ iand ha d(xi, xj)≤ i+ j
o each i, j ∈I. The e o e by ex e nal hype con exi y o E
(T
i
B(xi; i))(T
α
B(xα, α)) ∩E=T
α
B(xα, α)∩(A∩E)6=∅.
This leads o he ollowing.
Theo em 5.4. Le {Hi}be a descending chain o nonemp y ex e nally hype con ex subse s o a
bounded hype con ex space H. Then T
i
Hiis nonemp y and ex e nally hype con ex in H.
P oo . Theo em 5.1 assu es ha D=T
i
Hi6=∅.To see ha Dis ex e nally hype con ex le
{xα} ⊂ Hand { α} ⊂ Rsa is y d(xα, xβ)≤ α+ βand dis (xα, D)≤ α.Since His hype con ex
we know ha A=T
α
B(xα; α)6=∅.Also, since dis (xα, D)≤ αwe ha e dis (xα, Hi)≤ α o
each i, so, by ex e nal hype con exi y o Hi, we conclude A∩Hi6=∅ o each i. By Lemma 5.3
{A∩Hi}is a descending chain o nonemp y hype con ex subse s o H, so again by Theo em 5.1
T
i
(A∩Hi) = A∩D6=∅.
Rema k 5.5. Whe he Lemma 5.3 holds o weakly ex e nally hype con ex subse s is no known
ye , howe e i is a non e y complica ed exe cise o p o e ha Theo em 5.4 is ue o hese
subse s. An in e es ing open ques ion is whe he he in e sec ion o wo weakly ex e nally hype -
con ex subse s is s ill weakly ex e nally hype con ex. Such a p ope y is a e y impo an one
om a s uc u al poin o iew and i should help o show new ixed poin esul s o weakly ex-
e nally hype con ex subse s ha a e al eady known o admissible subse s.
One o he implica ions o Theo em 5.1 is he exis ence o hype con ex closu es. Indeed, le
Mbe a me ic space and conside he amily H(M) = {H;His hype con ex and M⊂H}.In
iew o wha we said p e iously, he amily H(M) is no emp y. Using Baillon’s esul , any de-
scending chain o elemen s o H(M) has a nonemp y in e sec ion. The e o e one may use Zo n’s
lemma which will insu e us o he exis ence o minimal elemen s. These minimal hype con ex
se s a e called hype con ex hulls. Isbell was among he i s o in es iga e he p ope ies o he
hype con ex hulls. In ac he was he i s one o gi e a conc e e cons uc ion o a hype con ex
hull in [24]. We will discuss his ideas in Sec ion 8.
I is clea ha hype con ex hulls a e no unique. Bu hey do enjoy some kind o uniqueness.
Indeed, we ha e:
P oposi ion 5.6. Le Mbe a me ic space. Assume ha H1and H2a e wo hype con ex hulls
o M. Then H1and H2a e isome ic.
P oo . Since H1and H2a e hype con ex, hen hey a e injec i e. So he e exis s a nonexpansi e
map T1:H1→H2such ha he es ic ion o T1 o Mis he iden i y map. Keep in mind ha
H1as well as H2con ains Misome ically. Fo he same eason, he e exis s also ano he
nonexpansi e map T2:H2→H1such ha he es ic ion o T2 o Mis he iden i y map. We
claim ha T1◦T2is he iden i y map o H2. Indeed, he map T1◦T2is de ined on H2in o H2.
I s es ic ion o Mis he iden i y map o M. So we ha e M⊂Fix(T1◦T2), whe e
Fix(T1◦T2) = {x∈H1;T1◦T2(x) = x}.
18 R. ESP´
INOLA AND M. A. KHAMSI
In he nex sec ion, we will show ha i Tis nonexpansi e, hen Fix(T) is hype con ex (see
Theo em 6.1), so, Since T1◦T2is nonexpansi e, Fix(T1◦T2) is hype con ex and con ains M.
The minimali y o H2implies
Fix(T1◦T2) = H2,
which comple es he p oo o ou claim. A simila a gumen will show ha T2◦T1is he iden i y
map o H1. So T1and T2a e in e se om each o he and a e nonexpansi e. The e o e bo h a e
isome ic maps.
Rema k 5.7. Though hype con ex hulls a e no unique, he p e ious p oposi ion shows ha up
o an isome y hey a e indeed unique. I is qui e an amazing esul . F om now on, we will deno e
he hype con ex hull o Mby h(M). Recall ha i Mis a subse o a hype con ex se H, hen
he e exis s a hype con ex hull h(M)such ha M⊂h(M)⊂H.
Isbell, in his s udy o he hype con ex hulls, showed ha i Mis compac hen h(M) is also
compac . As a gene aliza ion o his esul , we discuss nex some ideas de eloped by he i s
au ho and Lpez in [14, 16]. Since hei wo k in ol es measu e o noncompac ness, le us i s
gi e some de ini ions. Fo a wide ea men o hese de ini ions he eade is e e eed o [2].
De ini ion 5.8. Le Mbe a me ic space and le B(M)deno e he collec ion o nonemp y,
bounded subse s o M. Then:
(i) The Ku a owski measu e o noncompac ness α:B(M)→[0,∞)is de ined by
α(A) = in (ε > 0; A⊂
i=n
[
i=1
Aiwi h Ai∈ B(M)and diam(Ai)≤ε).
(ii) The Hausdo (o ball) measu e o noncompac ness χ:B(M)→[0,∞)is de ined by
χ(A) = in ( > 0; A⊂
i=n
[
i=1
B(xi, )wi h xi∈M).
These wo measu es a e e y much ela ed o each o he and o compac ness. Indeed, he
ollowing classical p ope ies a e well known.
(1) Fo any A∈ B(M), we ha e 0 ≤α(A)≤δ(A) = diam(A).
(2) Fo any A∈ B(M), we ha e α(A) = 0 i and only i Ais p ecompac .
(3) Fo any A∈ B(M) and B∈ B(M), we ha e α(A∪B) = max{α(A), α(B)}.
(4) Fo any A∈ B(M), we ha e χ(A)≤α(A)≤2χ(A).
(5) I (Ai)i∈Iis a dec easing chain o closed bounded se s such ha in i∈Iα(Ai) = 0, hen
T
i∈I
Aiis no emp y and is compac , i.e. αT
i∈I
Ai= 0.
In hype con ex me ic spaces, he wo measu es beha e nicely. Indeed, we ha e:
P oposi ion 5.9. Le Hbe a hype con ex me ic space and Abe a bounded subse o H. Then
we ha e α(A) = 2χ(A).
P oo . F om (4) i is enough o p o e ha 2χ(A)≤α(A).Le ε>χ(A). Then he e exis
A1, ..., Ansubse s o Asuch ha A=S1≤i≤nAiwi h diam(Ai)≤ε, o i= 1, ..., n. F om he
hype con exi y o H o any i∈ {1, ..., n}, he e exis s hi∈Hsuch ha Ai⊂Bhi,ε
2.Hence
INTRODUCTION TO HYPERCONVEX SPACES 19
A⊂[
1≤i≤n
Bhi,ε
2,which gi es α(A)≤ε
2.Hence α(A)≤χ(A)
2which comple es he p oo o
ou claim.
The ollowing echnical esul will be needed la e on. No e ha his esul may be seen as an
adap a ion o he classical A zel -Ascoli Theo em.
Lemma 5.10. Le Mbe a me ic space. Conside he space λ[a,b](M)o Lipschi zian eal- alued
unc ions de ined on Mwi h Lipschi z cons an less han λwi h alues in he in e al [a, b]. Then
we ha e
αλ[a,b](M)≤2λ χ(M).
P oo . Le ε0> χ(M). Wi hou loss o gene ali y, we may assume ha he e exis x1, ..., xn
in Msuch ha o any x∈M, he e exis s i∈ {1, ..., n}such ha d(x, xi)≤ε0. Since [a, b]
is compac , o any ε > 0, he e exis c1, ..., cmin [a, b] such ha o any c∈[a, b] he e exis s
i∈ {1, ..., m}such ha |c−ci| ≤ ε. Le ψ:{1, ..., n} → {1, ..., m}be an applica ion. De ine
λψ={ ∈λ[a,b](M); sup
1≤i≤n| (xi)−cψ(i)| ≤ ε}.
Though hese se s may be e en ually emp y, we s ill ha e ha
λ[a,b](M) = [
ψ∈{1,...,m}{1,...,n}
λψ.
Le , g ∈λψ. Fo any x∈M, he e exis s i∈ {1, ..., n}such ha d(x, xi)≤ε0. Then we ha e
 (x)−g(x)≤ (x)− (xi)+ (xi)−g(xi)+g(xi)−g(x),
which implies  (x)−g(x)≤λε0+ 2ε+λε0.Hence supx∈M (x)−g(x)≤2λε0+ 2ε. Since he
se {1, ..., m}{1,...,n}is ini e, we ge αλ[a,b](M)≤2λε0+ 2ε. So, om he a bi a iness o ε, we
ge αλ[a,b](M)≤2λε0,which clea ly implies αλ[a,b](M)≤2λ χ(M).
We will show in Sec ion 8 how Isbell cons uc ed o any bounded me ic space Ma hype -
con ex hull h(M) included in λ[0,δ](M), whe e λ= 1 and δis he diame e o M. So om he
abo e lemma, we deduce he ollowing esul .
Co olla y 5.11. Le Mbe any bounded me ic space and h(M)i s hype con ex hull. Then we
ha e χh(M)=χ(M)and αh(M)=α(M).
P oo . Since h(M) may be chosen such ha h(M)⊂λ[0,δ](M), whe e λ= 1 and δis he diame e
o M, i ollows αh(M)≤2χ(M).Using he p e ious lemma, χh(M)≤χ(M).Bu , since
h(M) con ains Misome ically, χ(M)≤χh(M),which clea ly implies he i s pa o he
conclusion. The second pa is a di ec consequence o P oposi ion 5.9.
6. Fixed poin p ope y and Hype con exi y
Sine [54] and Soa di [57] esul s a e a he o igin o he ecen in e es o hype con ex me ic
spaces. Bo h Sine and Soa di showed ha nonexpansi e mappings de ined on a bounded hy-
pe con ex me ic space ha e ixed poin s. Thei esul s we e s a ed in di e en con ex bu he
unde lying spaces a e simply hype con ex spaces. He e we will gi e he p oo based on Peno ’s
[49] o mula ion o Ki k’s ixed poin heo em.
20 R. ESP´
INOLA AND M. A. KHAMSI
Recall ha i T:M→Mis a map, hen x∈Mis a ixed poin o Ti T(x) = x.
Theo em 6.1. Le Hbe a bounded hype con ex me ic space. Any nonexpansi e map T:H→H
has a ixed poin . Mo eo e , he ixed poin se o T, Fix(T), is hype con ex.
P oo . Conside A(H) he amily o admissible subse s o H. Se
F={A∈ A(H); wi h A6=∅and T(A)⊂A}.
Ob iously, we ha e H∈ F. Since he in e sec ion o any amily o nonemp y elemen s o A(H)
is no emp y and belongs o A(H) (because o he hype con exi y o H), hen Fsa is ies Zo n’s
assump ions. So i has minimal elemen s. Le A0be one o hem. No e ha co (T(A0) = A0.
Indeed, since T(A0)⊂A0and co (T(A0)) is he smalles admissible se which con ains T(A0),
we ha e co (T(A0)) ⊂A0. Using his inclusion, we ge
T co T(A0)!⊂T(A0)⊂co T(A0).
This clea ly implies ha co T(A0)∈ F. The minimali y o A0will hen imply co (T(A0)) =
A0.Now no e ha C(A0) belongs o F. Indeed, we know ha C(A0) is no emp y and i is in
A(H) because
C(A0) =
x∈A0
Bx, R(A0).
Le x∈C(A0). Then we ha e A0⊂B(x, R(A0)).Since Tis nonexpansi e, we ge T(A0)⊂
B(T(x), R(A0)),which implies
A0= co T(A0)⊂BT(x), R(A0).
Hence T(x)∈C(A0). In o he wo ds, C(A0) is in a ian unde he ac ion o T. So we ha e
C(A0)∈ F. Ou claim is he e o e p o ed. The minimali y o A0will hen imply A0=C(A0).
Bu we ha e seen ha in hype con ex me ic spaces his is no possible o subse s wi h mo e
han one poin . This o ces A0 o ha e one poin which is a ixed poin o T. In o de o inish
he p oo o ou heo em, we need o show ha Fix(T) is hype con ex. Le {xi}i∈Ibe a collec ion
o poin s in Fix(T) such ha
d(xi, xj)≤ i+ j, o any i, j ∈I,
o some posi i e numbe s { i}i∈I. Se H0=
i∈I
B(xi, i). The hype con exi y o Himplies
ha H0is no emp y. Since he cen e s a e in Fix(T) and Tis nonexpansi e, hen we ha e
T(H0)⊂H0. Mo eo e H0is a bounded hype con ex me ic space, so he abo e p oo implies
ha Thas a ixed poin in H0, which implies
Fix(T) "
i∈I
B(xi, i)#6=∅.
This comple es he p oo o ou heo em.
This esul is qui e amazing. Indeed, i we ansla e i in o he hype con ex Banach space l∞,
we ha e, o example, ha any nonexpansi e mapping which lea es a ball in a ian has a ixed
poin . No e ha his space is qui e bad om a geome ical poin o iew. I is also a uni e sal
space o sepa able Banach spaces. In o he wo ds, any sepa able Banach space si s inside l∞
isome ically. So o he classical ixed poin p ope y his space is e y bad. And wi h his
INTRODUCTION TO HYPERCONVEX SPACES 21
heo em, we ha e a posi i e ixed poin esul .
No e ha since Fix(T) is hype con ex, hen any commu ing nonexpansi e maps Ti,i=
1,2, ..., n, de ined on a bounded hype con ex se H, ha e a common ixed poin . Mo eo e
hei common ixed poin se Fix(T1)∩Fix(T2)∩···∩Fix(Tn) is hype con ex.
Combining hese esul s wi h Baillon’s heo em, we ge he ollowing:
Theo em 6.2. Le Hbe a bounded hype con ex me ic space. Any commu ing amily o nonex-
pansi e maps {Ti}i∈I, wi h Ti:H→H, has a common ixed poin . Mo eo e , he common ixed
poin se
i∈I
Fix(Ti)is hype con ex.
P oo . Le Γ = 2I={β;β⊂I}. I is ob ious ha Γ is downwa d di ec ed ( he o de on Γ is he
se inclusion). Ou p e ious heo em implies ha o e e y β∈Γ, he se Fβo common ixed
poin se o he mappings Ti, i ∈β, is nonemp y and hype con ex. Clea ly he amily (Fβ)β∈Γ
is dec easing. Using Baillon’s esul , we deduce ha
i∈I
Fix(Ti) =
β∈Γ
Fβ
is nonemp y and hype con ex. The p oo is he e o e comple e.
Rema k 6.3. Baillon asked whe he boundedness may be elaxed. He p ecisely asked whe he
he conclusion holds i he nonexpansi e map has a bounded o bi . In he classical Ki k’s ixed
poin heo em, ha ing a bounded o bi implies he exis ence o a ixed poin . P us answe ed his
ques ion in he nega i e. Indeed, conside he hype con ex Banach space H=l∞and he map
T:H→Hde ined by
T(xn)= (1 + lim
Uxn, x1, x2, ...)
whe e Uis a non i ial ul a il e on he se o posi i e in ege s. We may also ake a Banach
limi ins ead o a limi o e an ul a il e . The map Tis an isome y and has no ixed poin . On
he o he hand, we ha e
Tn(0) = (1,1, ..., 1,0,0, ...)
whe e he i s block o leng h nhas all i s en ies equal o 1 and 0 a e ha . So Thas bounded
o bi s. This p oblem has been s udied in [35, 36, 40].
Recen ly, we wonde ed whe he his esul holds o asymp o ically nonexpansi e mappings.
Recall ha a map Tis said o be asymp o ically nonexpansi e i
d(Tn(x), Tn(y)) ≤λnd(x, y)
and lim
nλn= 1. This ques ion is ill unknown. Bu a pa ial posi i e answe is known o ap-
p oxima e ixed poin s. Be o e we s a e his esul , ecall ha i T:H→His a map, hen
x∈His an ε- ixed poin i d(x, T (x)) ≤εwhe e ε≥0. The se o ε- ixed poin s o Tis deno ed
by Fixε(T). Sine [55] ob ained he ollowing wonde ul esul :
Theo em 6.4. Le Hbe a bounded hype con ex me ic space and T:H→Ha nonexpansi e
map. Fo any ε > 0, Fixε(T)is no emp y and is hype con ex.

22 R. ESP´
INOLA AND M. A. KHAMSI
P oo . Fixε(T) is nonemp y since Thas ixed poin . Le {xα}α∈Γbe poin s in Fixε(T) and
{ α}α∈Γbe posi i e numbe s such ha d(xα, xβ)≤ α+ β o any α, β ∈Γ. Se
J=
α∈Γ
Bxα, α.
We know ha Jis nonemp y as a subse o H. We wish o show ha J∩Fixε(T) is no emp y. Le
x∈J. Then T(x)∈J+εbecause d(Tx, xα)≤d(Tx, Txα) + d(Txα, xα)≤d(x, xα) + ε≤ α+ε
o any α. Using Theo em 4.12, he e exis s a nonexpansi e e ac ion Π : J+ε→Jwhich
is ε-cons an . The map R:J→Jde ined by R(x) = Π ◦T(x) is nonexpansi e. Since Jis a
bounded hype con ex me ic space, hen Rhas a ixed poin x0∈J. Since Π is ε-cons an , we
ge
dx0, T(x0)=dΠ◦T(x0), T(x0)≤ε
which implies ha J∩Fixε(T)6=∅. The p oo is he e o e comple e.
Now we a e eady o s a e he ollowing unpublished esul .
Theo em 6.5. Le Hbe a bounded hype con ex me ic space and T:H→Hbe asymp o ically
nonexpansi e. Fo any ε > 0, Fixε(T)is no emp y, in o he wo ds we ha e
in
x∈Hd(x, T(x)) = 0.
P oo . Using he con exi y shown in Rema k 4.9, we de ine
Tn(x) = 1−1
λna⊕1
λn
Tn(x)
whe e ais a ixed poin in H,λnis he Lipschi z cons an o Tn. The maps Tna e nonexpansi e.
Be o e we p oceed wi h he p oo we need o de ine he “ul apowe ” o H. Conside he ca esian
p oduc H=Q
n≥1
H, and le Ube a non i ial ul a il e on he na u al numbe s. De ine he
equi alence ela ion ∼on Hby (xn)∼(yn) i and only i limUd(xn, yn) = 0.The limi o e U
exis s since His bounded. Then we conside he quo ien se e
H. An elemen ˜x∈e
His a subse
o H. I (xn)∈˜x, hen (yn)∈˜xi and only i lim
Ud(xn, yn) = 0. On e
Hde ine he me ic ˜
dby
˜
d(˜x, ˜y) = limUd(xn, yn) whe e (xn) ( esp. (yn)) is any elemen in ˜x( esp. ˜y). I is easy o see
ha e
Hendowed wi h he dis ance ˜
dhas many nice p ope ies simila o he linea ul apowe o
a Banach space. De ine he ope a o s ˆ
Tand ˜
Tby
ˆ
T(˜x) = ˆ
T(g
(xn)) = ^
(Tn(xn)) and ˜
T(˜x) = ˜
T(g
(xn)) = ^
(T(xn)).
Since Tis asymp o ically nonexpansi e, he ope a o ˆ
Tis nonexpansi e. Mo eo e we ha e
ˆ
T(g
(xn)) = ^
(Tn(xn)).Since Tnis nonexpansi e, i has a ixed poin xn. The poin ˜x=g
(xn) is
a ixed poin o ˆ
T. Hence he ixed poin se Fix( ˆ
T) is a nonemp y subse o e
H. Since he wo
ope a o s ˆ
Tand ˜
Tcommu e, hen ˜
Tlea es in a ian he se Fix( ˆ
T). I is easy o show ha ˜
T
es ic ed o Fix( ˆ
T) is in ac an isome y (in pa icula i is nonexpansi e). Fix ε > 0. Le
˜xi∈Fix( ˆ
T), i= 1, ..., N. Se
εn= max
1≤i≤Ndxn(i), Tn(xn(i))
o n≥1, whe e ˜xi=^
xn(i). Se
Hn={x∈H;d(x, Tn(x)) ≤εn}.
INTRODUCTION TO HYPERCONVEX SPACES 23
Hence xn(i)∈Hn, o i= 1, ..., N and any n≥1. Since Tnis nonexpansi e, Theo em 6.4
implies ha Hnis hype con ex. The e o e, he e exis zn(i) = εxn(1) ⊕(1 −ε)xn(i)∈Hn o
i= 1, ..., N. Conside , he poin ˜zi=^
zn(i), which we will deno e ε˜x1⊕(1 −ε)˜xi. Then we
ha e, ˜zi∈Fix( ˆ
T), and ˜
d(˜zi,˜zj)≤(1 −ε)d(˜xi,˜xj) o i, j = 2, ..., N. Back o ou maps ˆ
Tand ˜
T,
le ˜x∈Fix( ˆ
T), and w i e ˜x= ˜x1. Se
˜x2=ε˜x1⊕(1 −ε)˜
T˜x1.
Then ˜x2∈Fix( ˆ
T). By induc ion, we will cons uc a sequence ˜xno poin s in Fix( ˆ
T) de ined
by ˜xn+1 =ε˜x1⊕(1 −ε)˜
T˜xn.Fo any n<m, we ha e
d(˜xn,˜xm)≤(1 −ε)d ˜
T˜xn−1,˜
T˜xm−1!.
Since ˜
Tis nonexpansi e when es ic ed o Fix( ˆ
T), we ge
d(˜xn,˜xm)≤(1 −ε)d˜xn−1,˜xm−1.
This clea ly implies ha he sequence (˜xn) is a Cauchy sequence. Hence i con e ges o ˜ω∈
Fix( ˆ
T). Mo eo e we ha e
d˜ω, ˜
T(˜ω)= lim
n→∞ d˜xn+1,˜
T(˜xn)≤εlim
n→∞ d˜x1,˜
T(˜xn)
I we se δ= diame e (H), we ge d(˜ω, ˜
T(˜ω)) ≤εδ, so o any ε > 0 he e exis s ˜ωε∈Fix( ˆ
T)
such ha d˜ωε,˜
T(˜ωε)≤ε. Classical a gumen implies ha o any ε > 0 he e exis s xε∈H
such ha dxε, T(xε)≤εwhich comple es he p oo o Theo em 6.5.
7. Topological ixed poin heo ems and Hype con exi y
Ano he impo an b anch o ixed poin heo y is ha one o med by hose esul s in which
opological-like condi ions a e conside ed. We may hink o he well-known Schaude heo em
as he s a ing poin o his b anch. This heo em s a es ha any mapping de ined om a
nonemp y compac and con ex subse o a Banach space in o i sel mus ha e a ixed poin . An
easy imp o emen o his heo em is ob ained when he compac ness condi ion is imposed on he
mapping ins ead o on i s domain. New achie emen s came when he compac ness condi ion on
he mapping was ea ed in mo e gene al e ms. Le Mbe a me ic space and le B(M) be he
collec ion o nonemp y, and bounded subse s o M, hen a mapping γ:B(M)→[0,+∞) is called
ameasu e o noncompac ness i i sa is ies he ollowing condi ions:
(1) γ(A) = 0 i and only i Ais p ecompac .
(2) γ(A) = γ(A) o any A∈ B(M).
(3) γ(A∪B) = max{γ(A), γ(B)} o any A, B ∈ B(M).
O cou se, as i was announced in Sec ion 5, he mappings αand χgi en by De ini ion 5.8 a e
measu es o noncompac ness. A new kind o mapping a ises na u ally.
De ini ion 7.1. Le Mbe a me ic space and D⊆M. A mapping T:D→Mis said o be a
γ-condensing (o condensing ela i e o γ) mapping i Tis con inuous and i o each bounded
A⊆D, o which γ(A)>0,γ(T(A)) < γ(A).
24 R. ESP´
INOLA AND M. A. KHAMSI
A de ailed s udied o hese mappings may be ound in [2]. I is easy o see, howe e , ha
any compac mapping is condensing ela i e o any measu e γ. The also well-known Da bo-
Sado skii’s heo em [52] s a es ha i γis a measu e o noncompac ness de ined on a no med
space such ha γ(B) = γ(con (B)) o any nonemp y and bounded subse o he no med space,
and Tis a γ-condensing mapping om a nonemp y bounded closed and con ex subse o he
no med space in o i sel , hen Thas a ixed poin . The condi ion γ(B) = γ(con (B)) o in-
a iance when con ex hull is conside ed is undamen al o his esul and o he ela ed which
hype con ex coun e pa s will seen below. Measu es o Ku a owski and Hausdo s udied in
Sec ion 5 a e among hose ha sa is y he abo e condi ion. E en mo e, Co olla y 5.11 says ha
bo h measu es sa is y an equi alen condi ion o hype con ex spaces. We will make use o i la e
in his sec ion. Su p isingly, as i was no ed in [40], a hype con ex e sion o Da bo-Sado skii
heo em does no equi e o ha co olla y.
Theo em 7.2. Le Hbe a bounded hype con ex space and T:H→Haα-condensing mapping.
Then Thas a ixed poin .
P oo . F om Sec ion 3 we may assume ha His a bounded closed subse o a Banach space. We
also know ha he e exis s a nonexpansi e e ac ion R: cl-con (H)→H(whe e cl-con (H)
deno es he closed con ex hull o H). Then T◦R: cl-con (H)→H, and i A⊆cl-con (H)
sa is ies ha α(A)>0 hen ei he α(R(A)) = 0 o
α(T◦R(A)) < α(R(A)) ≤α(A).
In ei he case α(T◦R(A)) < α(A) so T◦Ris also α-condensing. Now Da bo-Sado skii heo em
implies he exis ence o a ixed poin x o T◦R. I is easy o see ha i mus also be a ixed
poin o T.
Rema k 7.3. No ice ha αmay be eplaced by any measu e o noncompac ness o which Da bo-
Sado skii heo em holds.
The classes o condensing ope a o s de ined ela i e o dis inc measu e o noncompac ness
a e no equal in gene al, bu hey ne e heless sha e a numbe o gene al p ope ies. Bea ing in
mind hese common p ope ies Sado skii [53] in oduced he concep o limi ope a o wi hou
using he no ion o measu e o noncompac ness. I Xis a linea space and Dis a subse o M,
hen a con inuous ope a o T:D→Xis called a limi compac o ul ima ely compac ope a o
i cl-con (T(B∩D)) = B, o B⊆X, implies ha Bis compac . The concep o hype con ex
hull will help us o de ine limi compac ope a o s in hype con ex spaces.
De ini ion 7.4. Le Dbe a subse o a hype con ex me ic space H. Gi en an ope a o T:D−→
Hwe will say ha (Tα)is a ans ini e sequence associa ed o Ton Di
(1) T0=hT(D)
(2) Tα=hT(D∩Tα−1)i α−1exis s,
(3) Tα=Tβ<α Tβ, i α−1does no exis ,
whe e Tαis a hype con ex hull o T(D)o T(D∩Tα−1)so ha he sequence (Tα)is noninc easing.
Rema k 7.5. I is easy o deduce om he p ope ies o he hype con ex hull ha gi en D,M
and as in he p e ious de ini ion, he e always exis s a ans ini e sequence associa ed o Ton D.
The p oo o he ollowing lemma is a he easy, we will omi i .
INTRODUCTION TO HYPERCONVEX SPACES 25
Lemma 7.6. I (Tα)is a ans ini e sequence associa ed o Ton D, hen:
(1) Each Tαis closed.
(2) T(D∩Tα)⊆Tα+1 o all α.
(3) I β < α, hen Tα⊆Tβ.
(4) T(D∩Tα)⊆Tα o all α.
(5) The e exis s an o dinal numbe ηsuch ha Tα=Tη o all α≥η.
Hence we ob ain he ollowing co olla y.
Co olla y 7.7. Le (Tα)be a ans ini e sequence associa ed o Ton H(hype con ex), and
suppose His bounded. Then he se Tηgi en by Lemma 7.6 is nonemp y.
P oo . I su ices o show ha o any o dinal numbe α,Tα6=∅. We p oceed by ans ini e
induc ion. The case α= 0 is i ial since T(D)⊆T0, and he e o e T0is nonemp y. Now we
ha e o conside he ollowing wo cases: he o dinal αhas a p edecesso , in which case, om he
induc i e hypo hesis, Tα−1is nonemp y and hence, Tα=h(T(M∩Tα−1)) is nonemp y, o he
o dinal αhas no p edecesso , in which case Tα=Tβ<α Tβ. By using he induc i e hypo hesis
and Baillon’s in e sec ion esul (Theo em 5.1), i ollows ha Tαis nonemp y.
This co olla y allows us o de ine he concep o limi compac ope a o in hype con ex spaces.
De ini ion 7.8. We say ha T∞(D)is an ul ima e ange o he ope a o Ton he se Hi
i is he limi se o a ans ini e sequence associa ed o Ton D. The ope a o Tis said o be
ul ima ely compac (o limi compac ) i is con inuous and he e exis s an ul ima e ange T∞(D)
o Tsuch ha T(D∩T∞(D)) is ela i ely compac on H.
We o e he ollowing lemma wi hou p oo , o de ails see [16].
Lemma 7.9. The ollowing p ope ies hold:
(1) T∞(D) = hT(D∩T∞(D)).
(2) I D1⊆D, hen T:D1→Hhas an ul ima e ange T∞(D1)con ained in T∞(D).
(3) I Tis ul ima ely compac on Dand D1⊆D,Tis ul ima ely compac on D1.
(4) The ope a o T:D→His ul ima ely compac i and only i T∞(D)is compac .
(5) The ope a o T:D→His ul ima ely compac i and only i o any B⊂H, he equali y
h( (B∩D)) = Bimplies ha Bis compac .
The ollowing heo em s a es he ela ion on α-condensing and limi compac mappings.
Theo em 7.10. Le Hbe a bounded hype con ex se , and suppose D⊆His closed. I T:D→H
is α-condensing, hen Tis ul ima ely compac on D.
P oo . Le T∞(D) be any ul ima e ange o Ton D. By he p e ious lemma hT(D∩T∞(D))=
T∞(D).F om his, hT(D∩T∞(D))⊇D∩T∞(D).Since αis mono onous, we ob ain αh(T(D∩
T∞(D))≥α(D∩T∞(D)).Bu , om Co olla y 5.11, αh(T(D∩T∞(D))=α(T(D∩T∞(D))).
And hence αT(D∩T∞(D))≥α(D∩T∞(D)).Bea ing in mind ha Tis condensing we con-
clude D∩T∞(D) is ela i ely compac . The e o e Tis ul ima ely compac on D.
The ollowing heo em s a es he exis ence o ixed poin o limi compac mappings, so, in
iew o he p e ious heo em, i may be unde s ood as an ex ension o Theo em 7.2.
Theo em 7.11. Le Hbe a bounded hype con ex me ic space and T:H→Ha limi compac
mapping on H. Then Thas a ixed poin in H.
32 R. ESP´
INOLA AND M. A. KHAMSI
Since d(˜
T(x0),˜
T(x)) = d(y0, T(x)) ≤d(x, x0) we conclude ha (D∪{x0},˜
T)∈F, con adic ing
he maximali y o (D, T).The e o e D=H. To conclude he p oo , we need o show ha Fix(T∗)
is hype con ex. This is a di ec consequence o Theo em 6.1 applied o T.
Rema k 9.5. Due o i s impo ance in di e en b anches o ma hema ics, selec ion p oblems
ha e been widely s udied along he las i y yea s. The p oblem is usually as ollows: gi en a
ce ain mul i alued mapping o be able o ind an uni alued selec ion o i wi h ce ain p op-
e ies as, o ins ance, con inui y o measu abili y. Theo em 9.1 is e y su p ising since i is
no common a all o be able o gua an ee ha a nonexpansi e mul i alued mapping admi s a
nonexpansi e selec ion, in ac his seems o be a qui e cha ac e is ic ac om he hype con ex
geome y. Hence o h one o he mos challenging open p oblems in hype con ex me ic spaces is
whe he his heo em may be imp o ed o no in he sense ha whe he E(H)may be eplaced by
a wide class o subse s o Mo no . Mo e speci ically he mos na u al ques ion a his momen
is whe he W(H), he class o nonemp y bounded weakly ex e nally hype con ex subse s, could
eplace E(H)in Theo em 9.1. Coun e examples a e no known o he case H(H). The eade
will ind a la ge collec ion o esul s and e e ences on mul i alued selec ion p oblems in he ecen
book [50].
We inish his sec ion wi h wo applica ion o Theo em 9.1. Fi s we show ha he amily o
all bounded λ-lipschi zian unc ions o a hype con ex space Min o i sel is i sel hype con ex
and second we will s udy a bes app oxima ion p oblem in hype con ex spaces.
Le and gbe wo bounded λ-lipschi zian unc ions o a hype con ex space Min o i sel , we
de ine he dis ance be ween hem in he usual way, ha is, i , g :M→M, se
d( , g) = sup
x∈M
d( (x), g(x)).
Theo em 9.6. Le Mbe hype con ex and o λ > 0le Fλdeno e he amily o all bounded
λ-lipschi zian unc ions o Min o M. Then Fλis i sel a hype con ex space.
P oo . Suppose { α} ⊂ Fλand { α} ⊂ Rsa is y d( α, β)≤ α+ β.Then o each x∈M
d( α(x), α(x)) ≤ α+ β,so in iew o he hype con exi y o M
J(x) =
α
B( α(x), α)6=∅
We show ha dH(J(x), J(y)) ≤λd(x, y) o each x, y ∈M. To see his i clea ly su ices o show
ha J(x)⊂Nλd(x,y)(J(y)). o each x, y ∈M. Howe e i z∈J(x) hen o each α
d(z, α(y)) ≤d(z, α(x)) + d( α(x), α(y))
≤d(z, α(x)) + λd(x, y)
≤ α+λd(x, y).
Using Sine’s Lemma (Lemma 4.10) we now ha e
z∈
α
B( α(y), α+λd(x, y)) = Nλd(x,y)(J(y))
In iew o Theo em 9.1 i is possible o selec (x)∈J(x) o each x∈Mso ha ∈Fλ.Since
∈T
α
B( α, α),Fλis hype con ex.
This leads o he ollowing.

INTRODUCTION TO HYPERCONVEX SPACES 33
Co olla y 9.7. Le Mbe a bounded hype con ex me ic space and le ∈ F1.Then he amily
R={ ∈ F1: (M)⊂Fix( )}
is a nonexpansi e e ac o F1.
P oo . The mapping T :F1→ F1de ined ia he o mula T (g) = ◦gis nonexpansi e and
has a nonemp y ixed poin se Fix(T ) which is hype con ex. Theo em 4.4 will hen imply ha
Fix(T ) is a nonexpansi e e ac o F1. Bu ∈Fix(T ) i and only i ∈R.
One o he mos impo an concep s in app oxima ion heo y is ha one o me ic p ojec ion.
In his las pa o he sec ion we s udy he p oblem o inding nonexpansi e selec ions o he
me ic p ojec ion. Le us in oduce some de ini ions i s . Recall ha he concep o p oximinali y
was in oduced in De ini ion 3.7.
De ini ion 9.8. Le Mbe a me ic space and Aa p oximinal subse o M, hen he mapping
R:M→2Ade ined as
R(x) = B(x, dis (x, A)) ∩A
o e e y x∈Mis called he me ic p ojec ion on A( ela i e o M).
No ice ha he p oximinali y o Agua an ees ha R(x)6=∅ o all x∈M. We will also deal
wi h he ollowing concep .
De ini ion 9.9. A subse Ao a me ic space Mis said o be a p oximinal nonexpansi e e ac
o Mi he e exis s a nonexpansi e selec ion o he me ic p ojec ion on A, i.e. i he e exis s a
nonexpansi e e ac ion :M→Asuch ha (x)∈B(x, dis (x, A)) ∩A o each x∈M.
The i s one in aking up he p oblem o cha ac e izing p oximinal nonexpansi e e ac s in
hype con ex me ic spaces was Sine [56]. The ollowing heo em was s a ed in [56] o admissible
subse s, we adap Sine’s p oo o ex e nally hype con ex subse s.
Theo em 9.10. Le Ebe a nonemp y ex e nally hype con ex subse o a me ic space H. Then
Eis a p oximinal nonexpansi e e ac o H.
P oo . Fo each x∈Hwe de ine A(x) = B(x, dis (x, E)) ∩Eand
C(x) = {A(y) + d(x, y) : y∈H}.
Since A(x) is one o he se s ha de ine C(x), he inclusion C(x)⊆A(x) is clea . We will show
ha C:H→2Eis a nonexpansi e mul i alued mapping which alues a e nonemp y ex e nally
hype con ex subse s o H. Due o he ex e nally hype con exi y o Ei will be enough o p o e,
in o de o s a e he nonemp yness o A(x), ha
B(y1,dis (y1, E) + d(x, y1)) ∩B(y2,dis (y2, E) + d(x, y2)) 6=∅
o each y1and y2in H. Bu xbelongs o bo h o hese se s, so C(x)6=∅. Addi ionally, he
ex e nally hype con exi y o C(x) ollows as an easy consequence o Lemma 5.3. Le us see now
ha Cis nonexpansi e, o his pick uand in H. We ha e o show ha C(u)⊆C( ) + d(u, ),
o equi alen ly
z∈H
(A(z) + d(u, z)) ⊆
z∈H
(A(z) + d(z, ) + d(u, )),
which is clea i one ecalls ha d(u, z)≤d(u, ) + d( , z). Now he heo em ollows as an appli-
ca ion o Theo em 9.1.
34 R. ESP´
INOLA AND M. A. KHAMSI
This p oblem was aken up again in [15] wi h some imp o emen s o he abo e heo em. We
s a e hese esul s wi hou p oo s, which may be ound in [15].
Theo em 9.11. A compac subse Eo a hype con ex me ic space His a p oximinal nonex-
pansi e e ac o Hi and only i Eis weakly ex e nally hype con ex ( ela i e o H).
The si ua ion o he noncompac case has no been sol ed ye al hough he ollowing heo em
s a es a pa ial esul ha ge s e y close o comple ely emo e he compac condi ion on Theo em
9.11.
Theo em 9.12. Le Ebe a weakly ex e nally hype con ex subse o a hype con ex me ic space
H. Then gi en any ε > 0 he e is a nonexpansi e e ac ion Rεo Hon o Ewi h he p ope y
ha gi en any u∈H E he e exis s x∈Hsuch ha d(u, x)≤εand d(x, Rε(x)) = dis (x, E).
Mo eo e i in (E)6=∅ hen Rεmay be chosen so ha Rε(H E)⊂∂E.
Sine made used o Theo em 9.10 in o de o ob ain ce ain Ky Fan [17] ype heo ems o hy-
pe con ex spaces. E en hough Theo em 9.12 does no qui e sol e he “nonexpansi e p oximinal
e ac ” p oblem i s ill enjoys enough good p ope ies how o lead o imp o ed new e sions o
Ky Fan ype esul s gi en by Sine. We conclude his sec ion wi h hese ixed poin esul s.
The i s esul we will see is a opological one, as hose seen in Sec ion 7, wi h bounda y
condi ion. In ac i ex ends Da bo-Sado skii Theo em.
Theo em 9.13. Suppose Eis a bounded weakly ex e nally hype con ex subse o a hype con ex
space Hwi h nonemp y in e io , and le T:E→Hbe a uni o mly con inuous condensing
mapping o which T(∂E)⊂E. Then Thas a ixed poin .
P oo . Le ε > 0 and choose ε0≤εso ha d(u, )≤ε0⇒d(T(u), T ( )) ≤ε. Now le Rε0
be he nonexpansi e e ac ion assu ed by Theo em 9.12.I is easy o see ha he mapping
Rε0◦T:E→Eis condensing, and since Eis hype con ex Rε0◦Thas a ixed poin , say xε∈
D. I T(xε)∈E, hen Rε0◦T(xε) = T(xε) = xε. I T(xε)/∈E, hen he e exis s y∈∂E such
ha d(xε, y)≤ε0.In his case (since T(y)∈E) we ha e
d(y, T (y)) ≤d(y, xε) + d(xε, T (y))
≤ε+d(Rε0◦T(xε), Rε0◦T(y))
≤ε+d(T(xε), T (y))
≤2ε.
This p o es ha in {d(y, T (y)) : y∈E}= 0.Since Tis condensing i easily ollows ha Thas
a ixed poin in D.
The ollowing is an easy consequence o Theo em 9.12.
Theo em 9.14. Suppose Eis a bounded weakly ex e nally hype con ex subse o a hype con ex
me ic space H, and suppose T:E→His a nonexpansi e mapping o which T(∂E)⊂E. Then
Thas a ixed poin .
P oo . Le Rbe a nonexpansi e e ac ion o Hon o E o which R(H E)⊆∂E. Then
R◦T:E→Eis nonexpansi e and has he same ixed poin se as T.
Ano he consequence o Theo em 9.12 is he ollowing heo em.
INTRODUCTION TO HYPERCONVEX SPACES 35
Theo em 9.15. Suppose Eis a bounded weakly ex e nally hype con ex subse o a hype con ex
me ic space H, and le T:E→Hbe a condensing mapping o which T(∂E)⊂E. Then Thas
a ixed poin .
Finally, since compac hype con ex spaces ha e he ixed poin p ope y o con inuous map-
pings (e.g., [31, 47]), Theo em 9.11 yields Ky Fan’s app oxima ion p inciple o compac weakly
ex e nally hype con ex se s.
Theo em 9.16. Le Eand Hsa is y he assump ions o Theo em 9.11 and suppose T:E→H
is a con inuous mapping. Then he e exis s x∈Esuch ha
d(x, T (x)) = in {d(y, T (x)) : y∈D}.
P oo . Le Rbe he e ac ion gi en by Theo em 9.11, hen R◦T:E→Eis a con inuous
mapping so i has a ixed poin in Ewhich con i ms he s a emen o he heo em.
Ky Fan’s ixed poin heo em o hype con ex me ic space is hence o h s a ed in he ollowing
way.
Co olla y 9.17. Le Eand Hsa is y he assump ions o Theo em 9.11 and suppose T:E→H
is a con inuous mapping such ha T(∂E)⊆E(i in e io o Eis emp y, assume T(E)⊆E).
Then Thas a ixed poin in E.
P oo . I is easy o see ha he ixed poin o R◦Tin he heo em mus be a ixed poin o T
unde he addi ional bounda y condi ion o he s a emen o he co olla y.
10. The KKM heo y in Hype con ex spaces
Among he esul s equi alen o he B ouwe ’s ixed poin heo em, he Theo em o Knas e -
Ku a owski-Mazu kiewicz (in sho KKM) occupies a special place. His o ically B ouwe ’s ixed
poin heo em ailed o impose i sel in he me ic se ing in compa ison wi h Ki k’s ixed poin
heo em. In ou opinion, his is due o he ac ha he i s heo em depends hea ily on he
con ex s uc u e o he se while he second one depends on a se heo e ical kind o con exi y.
Since hype con ex me ic spaces exhibi some kind o con exi y, i was na u al o in es iga e
B ouwe ’s ixed poin heo em in his se ing. This p oblem was i s s udied by he second
au ho in [31] and la e by o he s in [41, 47]. This sec ion is de o ed o p esen some o he
esul s appea ed in hese h ee wo ks. The li e a u e on KKM p inciple is qui e la ge, he eade
will ind a mo e p ecise and exhaus i e ea men o i in [22, 59] whe e KKM p inciple is mainly
de eloped in opological and nonlinea se ings.
Le Hbe a me ic space. A subse A⊂His called ini ely closed i o e e y x1, x2, ..., xn∈H,
he se co ({xi})∩Ais closed. I Ais closed hen ob iously i is also ini ely closed. Recall ha
a amily {Aα}α∈Γin 2His said o ha e he ini e in e sec ion p ope y i he in e sec ion o each
ini e sub amily is no emp y.
De ini ion 10.1. Le Hbe a me ic space and X⊂H. A mul i alued mapping G:X→2His
called a Knas e -Ku a owski-Mazu kiewicz map (in sho KKM-map) i
co ({x1, ..., xn})⊂[
1≤i≤n
G(xi)
o any x1, ..., xn∈X.
36 R. ESP´
INOLA AND M. A. KHAMSI
We ha e he ollowing esul :
Theo em 10.2. (KKM-maps p inciple) Le Hbe a hype con ex me ic space, and Xbe a
nonemp y subse o H. Le G:X→2Hbe a KKM-map such ha each G(x)is ini ely closed.
Then he amily {G(x); x∈X}has he ini e in e sec ion p ope y.
P oo . Assume no , i.e. he e exis x1, ..., xn∈Xsuch ha
i=n
T
i=1
G(xi) = ∅. Se L= co ({xi})
in H. Conside he hype con ex Banach space l∞(H) and se H∞= co (H) in l∞(H). Le
C= con (xi) in H∞. He e we conside he linea con ex hull. By Theo em 4.4, he e exis s
a nonexpansi e e ac ion :H∞→H. No e ha (C)⊂L. Ou assump ions imply ha
L∩G(xi) is closed o e e y i= 1,2, ..., n. Since T
i
G(xi)TL=∅ hen, o e e y c∈C, he e
exis s i0such ha (c) does no belong o L∩G(xi0). Hence dis ( (c), L ∩G(xi0)) >0 because
L∩G(xi0) is closed. The e o e, he unc ion
α(c) =
i=n
X
i=1
dis  (c), L ∩G(xi)
is no ze o o any c∈C. De ine he map F:C→Cby
F(c) = 1
α(c)
i=n
X
i=1
dis  (c), L ∩G(xi)xi.
Clea ly, Fis a con inuous map. Since Cis compac , hen B ouwe ’s heo em implies he exis ence
o a ixed poin c0o F, i.e. F(c0) = c0. Se I={i; dis ( (c0), L ∩G(xi)) 6= 0}.Clea ly we ha e
c0=1
α(c0)X
i∈I
dis  (c0), L ∩G(xi)xi.
The e o e, (c0)6∈ S
i∈I
G(xi) and (c0)∈co ({xi;i∈I}), con adic ing he assump ion co ({xi;i∈
I} ⊂ ∪i∈IG(xi).The p oo o Theo em 10.2 is he e o e comple e.
As an immedia e consequence, we ob ain he ollowing heo em.
Theo em 10.3. Le Hbe a hype con ex me ic space and X⊂Hbe a nonemp y subse . Le
G:X→2Hbe a KKM-map such ha G(x)is closed o any x∈Xand G(x0)is compac o
some x0∈X. Then we ha e
x∈X
G(x)6=∅.
No ice ha he compac ness assump ion o G(x0) may be a s onge one. We can s ill each
he conclusion i one in ol es an auxilia y mul i alued map and a sui able opology on H(such
as he ball opology o example).
Theo em 10.4. Le Hbe a hype con ex me ic space and X⊂Hbe a nonemp y subse . Le
G:X→2Hbe a KKM-map. Assume he e exis s a mul i alued map K:X→2Hsuch ha
G(x)⊂K(x) o e e y x∈Xand
x∈X
K(x) =
x∈X
G(x).
INTRODUCTION TO HYPERCONVEX SPACES 37
I he e exis s a opology τon Hsuch ha each K(x)is compac o τ, hen
x∈X
G(x)6=∅.
The p oo is ob ious.
The concep o KKM mapping was gene alized in [41] in he ollowing way.
De ini ion 10.5. Le Hbe a me ic space and X⊆H. A mul i alued mapping G:X→2H {∅}
is called a gene alized me ic KKM mapping (GMKKM) i o each ini e se {x1, ..., xn} ⊆ X,
he e exis s a se {y1, ..., yn}o poin s o H, no necessa ily all di e en , such ha o each subse
{yi1, ..., yik}o {y1, ..., yn}we ha e co {yij:j= 1, ...k} ⊆ ∪k
j=1G(xij).
I is easy o check ha KKM mappings a e gene alized KKM mappings while he con e se
is no ue, he in e es ed eade may consul [41, 59] o mo e abou his opic. The ollowing
heo em is he ex ension o Theo em 10.2 whe e gene alized me ic KKM mappings subs i u e
KKM mappings. The p oo , al hough mo e complica ed, ollows simila ideas o hose in he
p oo o Theo em 10.2 and we will omi i .
Theo em 10.6. (Gene alized me ic KKM p inciple) Le Hbe a hype con ex me ic space, and
Xbe a nonemp y subse o H. Suppose G:X→2H {∅} has ini ely closed alues. Then he
amily {G(x) : x∈X}has he ini e in e sec ion p ope y i and only i he mapping Gis a
gene alized me ic KKM mapping.
Ki k a al. exhibi a la ge numbe o consequences o his heo em in [41] (see also [40, 47, 59]).
These consequences ha e o do wi h Minimax inequali ies, ixed poin heo ems o mul i alued
mappings, saddle poin s, and Nash equilib ia. The ollowing heo em is among hese conse-
quences. No e ha his heo em is a mul i alued e sion o Ky Fan’s app oxima ion p inciple
al eady seen in he p e ious sec ion.
Theo em 10.7. Le Hbe a hype con ex space and Aa nonemp y admissible compac subse o
H. Suppose T:A→ A(H)is a mul i alued con inuous mapping. Then he e exis s x0∈Asuch
ha
dis (x0, T(x0)) = in
x∈Adis (x, T(x0)).
P oo . De ine he mapping G:A→2H {∅} by G(x) = {y∈A: dis (y, T(y)) ≤dis (x, T(y))}
o each x∈A. As Tis con inuous, G(x) is closed and nonemp y o each x∈A. We wan o
p o e ha Gis a KKM mapping. Suppose i is no , hen he e exis s a nonemp y and ini e subse
{x1, ..., xn}and y∈co ({xi:i= 1, ..., n}) such ha dis (xi, T(y)) <dis (y, T(y)) o i= 1, ..., n.
Le ε > 0 be such ha dis (xi, T(y)) ≤dis (y, T(y)) −ε o i= 1, ..., n. Le = dis (y, T(y)) −ε.
Then xi∈T(y) + o i= 1, ..., n. F om Lemma 4.10 T(y) + ∈ A(H), hus co {x1, ..., xn} ⊆
T(y)+ . This in u n implies y∈T(y)+ and hence dis (y, T(y)) ≤ = dis (y, T(y))−ε, which
is no possible by assump ion. The e o e Gmus be a KKM mapping.
No e ha ∩x∈AG(x)6=∅since Xis compac . Take x0∈ ∩x∈AG(x). Then i is clea ha
dis (x0, T(x0)) ≤dis (x, T(x0)) o all x∈A, which comple es he p oo o he heo em.
11. Lambda-Hype con exi y
Since he beginning i was known ha he Hilbe space `2 ails hype con exi y. By s udying
his case closely, he second au ho , Knaus , Nguyen and O’Neill [34] in oduced a p ope y e y

38 R. ESP´
INOLA AND M. A. KHAMSI
simila o hype con exi y, called λ-hype con exi y. The idea is o expand he adius o he gi en
balls by a uni o m ac o . Fo example, e e y pai wise in e sec ing collec ion o balls in `2has
non-emp y in e sec ion i he adius o he balls a e inc eased by he ac o √2. In ligh o his,
he ollowing de ini ion becomes na u al.
De ini ion 11.1. Le Mbe a me ic space and le λ≥1. We say ha he me ic space Mis
λ-hype con ex i o e e y non-emp y admissible se A∈ A(M), o any amily o closed balls
{B(xα, α)}α∈Λ, cen e ed a xα∈A o α∈Λ, he condi ion
d(xα, xβ)≤ α+ β o e e y α, β ∈Λ,
implies
A∩
α∈Λ
B(xα, λ α)6=∅.
Le λ(M)be he in imum o all cons an s λsuch ha Mis λ-hype con ex, and say ha λ(M)is
exac i Mis λ(M)-hype con ex.
G ¨unbaum [20] and o he au ho s ha e s udied a simila p ope y no in ol ing he unde lying
admissible se A. Bu i s in oduc ion becomes essen ial when we y o connec his concep o
he ixed poin p ope y ia he no mal s uc u e p ope y.
Le us ecall G ¨unbaum’s de ini ion: Fo a me ic space M, le he expansion cons an E(M) be
he in imum o all cons an s µsuch ha he ollowing holds: Whene e a collec ion {B(xα, α) :
α∈Λ}in e sec s pai wise, hen
α∈Λ
B(xα, µ · α)6=∅.
We say E(M) is exac , i he condi ion is e en sa is ied o µ=E(M).
T i ially, E(M)≤Λ(M) holds in me ically con ex spaces (see De ini ion 2.3). On he o he
hand, i Mis a wo elemen me ic space, hen E(M) = 1, while Λ(M) = 2, so bo h concep s do
no coincide in gene al.
Le us i s summa ize some basic p ope ies o λ-hype con ex me ic spaces, some o which
a e i ial, while he o he s can be easily de i ed om co esponding esul s abou expansion
cons an s:
Theo em 11.2. Le Mbe a me ic space.
1. Mis hype con ex i and only i i is 1-hype con ex.
2. E e y λ-hype con ex me ic space is comple e.
3. Re lexi e Banach spaces and dual Banach spaces a e 2-hype con ex.
4. The e is a subspace Xo `1which ails o be 2-hype con ex.
5. Hilbe space is √2-hype con ex.
We will inish his sec ion, and hence his chap e on hype con exi y, by s udying he con-
nec ion be ween λ-hype con exi y and he ixed poin p ope y. The heo em we will inish wi h
is based on a me ic gene aliza ion o Ki k’s ixed poin heo em es ablished in [30]. In o de
o s a e his gene aliza ion we need o know wha a uni o m no mal s uc u e on a me ic space is.
Le Mbe a me ic space and Fa amily o subse s o M. Then we say ha Fde ines a
con exi y s uc u e on Mi i con ains he closed balls and is s able by in e sec ion. Fo ins ance
A(M), he class o he admissible subse s o M, de ines a con exi y s uc u e on any me ic
INTRODUCTION TO HYPERCONVEX SPACES 39
space M. We say ha Fis a uni o m no mal s uc u e on Mi he e exis s c < 1 such ha
R(A)≤c·diam(A) o e e y A∈ F wi h diam(A)>0, whe e R(A) and diam(A) a e, espec-
i ely, he Chebyshe adius and diame e o Ade ined in Sec ion 3.
Now we may s a e he gene aliza ion o Ki k’s ixed poin heo em.
Theo em 11.3. Le Mbe a bounded comple e me ic space. I Mhas a uni o m no mal s uc-
u e hen i has he ixed poin p ope y o nonexpansi e mappings.
The connec ion be ween λ-hype con exi y and he ixed poin p ope y is gi ing by he ol-
lowing heo em.
Theo em 11.4. Le Mbe a bounded λ-hype con ex space. I λ < 2, hen any nonexpansi e
mapping T:M→Mhas a ixed poin .
P oo . Le Mbe a bounded λ-hype con ex space wi h λ < 2. Theo em 11.2 assu es ha M
is comple e, so om he p e ious heo em i su ices o p o e ha Mhas a uni o m no mal
s uc u e. The amily A(M) de ines a con exi y s uc u e on M, we will show ha A(M) is
ac ually a uni o m no mal s uc u e on M. Le A∈ A(M) wi h diam(A)>0. Fo each x∈Ale
B(x, x) deno e he ball cen e ed a xwi h cons an adius x=1
2diam(A). Then d(x, y)≤ x+ y
o e e y x, y ∈A. Since Mis λ-hype con ex we can ind
x0∈A∩(∩x∈AB(x, λ x)).
Thus we ha e d(x, x0)≤1
2λdiam(A) o e e y x∈A. I ollows ha R(A)≤1
2λdiam(A). Finally
since 1
2λ < 1 we ob ain ha A(M) is a uni o m no mal s uc u e on M, and hence he heo em
is p o ed.
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40 R. ESP´
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