Hous on Jou nal o Ma hema ics
c
2004 Uni e si y o Hous on
Volume 30, No. 1, 2004
ON A UNIFIED STUDY OF RELATIVE CHEBYSHEV RADII
AND HAUSDORFF MEASURES OF NONCOMPACTNESS
RAFAEL ESP´
INOLA1, ANDRZEJ WI´
SNICKI 2, AND JACEK WO´
SKO 2
Communica ed by Gilles Pisie
Abs ac . The pape is conce ned wi h he no ion o he Li schi z modu-
lus and i s ela ionship wi h bo h ela i e Chebyshe adii and Hausdo
measu es o noncompac ness.
1. In oduc ion
Th oughou he pape we shall use he ollowing no a ion: Xwill deno e a eal
no med space, Aa nonemp y, bounded subse o Xand Ga nonemp y subse o
X. As usual, B(x, ) will s and o he closed ball cen e ed a x∈Xwi h adius
> 0, diam A o he diame e o A, and dis (x, A) o he dis ance om x o A.
We shall also w i e
B(A, ) = {x∈X: dis (x, A)≤ },
dis (A, G) = in {dis (x, G) : x∈A},
d(A, G) = in { > 0 : A⊂B(G, )}
and
d(H, G) = in {d(F, G) : F∈ H}
i His a nonemp y amily o nonemp y bounded subse s o X.
Le us ecall ha he ela i e Chebyshe adius G(A) is gi en by
G(A) = in { > 0 : A⊂B(y, ) o some y∈G}
2000 Ma hema ics Subjec Classi ica ion. P ima y 46B20, 41A65, 47H09.
1Suppo ed by he p ojec PB96-1338-C02-01 om Jun a de Andalucia.
2Suppo ed in pa by he g an 2 PO3A 029 15 om he S a e Commi ee o Scien i ic
Resea ch o Poland.
245
246 RAFAEL ESP´
INOLA, ANDRZEJ WI´
SNICKI, AND JACEK WO´
SKO
and he ela i e Hausdo measu e o noncompac ness χG(A) by
χG(A) = in { > 0 : A⊂B(F, ) o some ini e se F⊂G}.
I G=Xwe shall abb e ia e (A) and χ(A) espec i ely. Finally, o ¯≥0,
Eε(A) = {y∈X:A⊂B(y, (A) + ε)},
Hε(A) = {F⊂X:A⊂B(F, χ (A) + ε) and Fis ini e}.
No e ha Hε(A)6=∅and Eε(A)6=∅ o e e y bounded A⊂Xand ε > 0.
The cen al heme o his pape is he concep o he Li schi z modulus, de ined
in [16] as ollows:
De ini ion 1.1. The Li schi z modulus o a no med space Xis he unc ion ˜κX(·)
de ined on [0,+∞) by
˜κX(d) = sup{k > 0 : ∃α∈(0,1) ∀x, y ∈X∀ > 0∃z∈X(kz−yk ≤ αd
∧B(x, )∩B(y, k )⊂B(z, ))}.
The o igins o his no ion come om [13], whe e sligh ly simila ideas we e
used o p o e a ce ain ixed poin heo em o uni o mly Lipschi zian mappings.
Al hough he de ini ion o he Li schi z modulus seems o be a li le a i icial, i is
based on e y na u al geome ic in ui ion (see Sec ion 2 o mo e de ails). Wi h
he help o his no ion we managed [16] o ex end he o mula
G(A) = (A) + dis E0(A), G¡
(p o ed in he case X=C(K) by Smi h and Wa d in [15], see also [7]), o a
wide class o spaces including c0as well as some subspaces o C(K).Mo eo e ,
he concep o he Li schi z modulus le us ea he no ions o ela i e Chebyshe
adii and ela i e Hausdo measu es o noncompac ness in a uni ied way, since
Theo em 1.2, s a ed below, implies ha i ˜κX(d) = 1 + d o all d≥0, hen
G(A) = (A) + lim
ε→0+dis (Eε(A), G)
and
χG(A) = χ(A) + lim
ε→0+d(Hε(A), G).
The ollowing heo em is a mino modi ica ion o he esul gi en in [16]:
Theo em 1.2. Le Aand Gbe nonemp y subse s o a no med space Xwi h A
nonsingle on and bounded. Then
G(A)≥ (A) ˜κX°d (A,G)
(A)±,
RELATIVE CHEBYSHEV RADII AND HAUSDORFF MEASURES 247
whe e
d (A, G) = (dis E0(A), G¡i E0(A)6=∅,
lim
ε→0+dis (Eε(A), G)i E0(A) = ∅.
I χ(A)6= 0, hen
χG(A)≥χ(A) ˜κX°dχ(A,G)
χ(A)±,
whe e
dχ(A, G) = (dH0(A), G¡i H0(A)6=∅,
lim
ε→0+d(Hε(A), G)i H0(A) = ∅.
In [6] we pu hese esul s in a mo e gene al amewo k and ga e he ollowing
cha ac e iza ion o C(K) and C0(Ω) spaces:
Theo em 1.3. Le Aand Gbe nonemp y subse s o a Banach space Xwi h A
bounded. The ollowing asse ions a e equi alen :
(1) G(A) = (A) + dis E0(A), G¡ o e e y Aand G, as abo e.
(2) G(A) = (A) + lim
ε→0+dis (Eε(A), G) o e e y Aand G, as abo e.
(3) Xis isome ic o C(K) o some compac Hausdo opological space K
o o C0(Ω) o some locally compac Hausdo space Ω.
(4) ˜κX(d) = 1 + d o all d≥0.
The p esen pape deals wi h he ollowing wo ques ions:
1) How “good” a e he es ima ions ob ained in Theo em 1.2?
2) Does he cha ac e iza ion o C(K) and C0(Ω) spaces gi en in Theo em 1.3
s ill hold i we eplace he no ion o ela i e Chebyshe adii by ela i e Hausdo
measu es o noncompac ness?
In Sec ion 2 we discuss some p ope ies o he Li schi z modulus. Wi h he use
o hese p ope ies we show in Sec ion 3 ha he es ima ion o G(A) gi en in
Theo em 1.2 is, in a sense, op imal. We ecall Theo em 1.3 in Sec ion 4 in o de
o s udy i s coun e pa o he ela i e Hausdo measu e o noncompac ness.
I clea ly ollows ha i Xis isome ic o C(K) o C0(Ω), hen
(?)χG(A) = χ(A) + lim
ε→0+d(Hε(A), G).
Howe e , a om s a ing an equi alen cha ac e izing esul we p o e ha o -
mula (?) does hold in he space `1which is well-known o be nonisome ic o he
spaces C(K) and C0(Ω). Thus, we disco e he a he su p ising ac ha he
no ions o ela i e Chebyshe adii and Hausdo measu es o noncompac ness
beha e in a di e en way in his p oblem.
Sec ion 5 con ains some ema ks conce ning ano he p oblem on ela i e Cheby-
she adii and ela i e Hausdo measu es o noncompac ness.
248 RAFAEL ESP´
INOLA, ANDRZEJ WI´
SNICKI, AND JACEK WO´
SKO
2. P ope ies o he Li schi z Modulus
I is easily seen ha in any no med space X
max {1, d −1} ≤ ˜κX(d)≤d+ 1
o all d≥0.The nex wo p oposi ions show ha we may ea he pa ame e d
in he de ini ion o he modulus o Li schi z as he dis ance be ween xand y.
P oposi ion 2.1. Fo all d≥0
˜κX(d) = sup{k > 0 : ∃α∈(0,1) ∀x, y ∈X(kx−yk ≤ d⇒
∃z∈X(kz−yk ≤ αd ∧B(x, 1) ∩B(y, k)⊂B(z, 1)))}.
P oo . Le us i s no ice ha we can ix = 1 in he de ini ion o ˜κX. Pu
d > 0 and ake k≥1 and α∈(0,1) such ha o e e y x, y ∈X, we ob ain
(2.1) kx−yk ≤ d⇒ ∃z∈X(kz−yk ≤ αd ∧B(x, 1) ∩B(y, k)⊂B(z, 1)) .
I is su icien o p o e ha k≤˜κX(d). Le x, y ∈Xwi h kx−yk> d. We
need o cons uc z∈Xsuch ha kz−yk ≤ αd ∧B(x, 1) ∩B(y, k)⊂B(z, 1).
Le y1be he elemen o he segmen [x, y] sa is ying he equali y kx−y1k=
d. The e o e he e exis s z1∈Xwi h he p ope y ha kz1−y1k ≤ αd and
B(x, 1) ∩B(y1, k)⊂B(z1,1).We now conside wo cases.
Assume i s ha ky1−yk ≤ (1−α)d. The e o e kz1−yk≤kz1−y1k+ky1−yk ≤
αd + (1 −α)d=dand hence using (2.1), we ob ain B(z1,1) ∩B(y, k)⊂B(z, 1)
o some zwi h kz−yk ≤ αd. Thus, since k≥1,
B(x, 1) ∩B(y, k) = B(x, 1) ∩B(y1, k)∩B(y, k)⊂B(z1,1) ∩B(y, k)⊂B(z, 1)
and we ha e he desi ed esul .
Assume now ha ky1−yk>(1 −α)d. The e o e ky1−yk ≤ n(1 −α)d o
some n∈Nand we can ind he elemen s y2, y3, ..., yno he segmen [y1, y] such
ha ky1−y2k= (1 −α)d,ky1−y3k= 2(1 −α)d, ..., ky1−ynk= (n−1)(1 −α)d
and kyn−yk ≤ (1 −α)d. Using simila a gumen s as be o e we ind he elemen s
z2, z3, ..., znsuch ha
B(x, 1) ∩B(y2, k)⊂B(z2,1),kz2−y2k ≤ αd,
B(x, 1) ∩B(y3, k)⊂B(z3,1),kz3−y3k ≤ αd,
......
B(x, 1) ∩B(yn, k)⊂B(zn,1),kzn−ynk ≤ αd.
Finally, B(x, 1) ∩B(y, k)⊂B(z, 1) o some zsa is ying kz−yk ≤ αd and he
p oo is comple e. £
RELATIVE CHEBYSHEV RADII AND HAUSDORFF MEASURES 249
P oposi ion 2.1 leads o he a o emen ioned desi ed esul .
P oposi ion 2.2. Fo all d≥0
˜κX(d) = sup{k > 0 : ∃α∈(0,1) ∀x, y ∈X(kx−yk=d⇒
∃z∈X(kz−yk ≤ αd ∧B(x, 1) ∩B(y, k)⊂B(z, 1)))}.
P oo . Fixing d > 0 and aking k > 0 and α0∈(0,1) such ha o e e y
x, y ∈X, we ob ain
kx−yk=d⇒ ∃z∈X(kz−yk ≤ α0d∧B(x, 1) ∩B(y, k)⊂B(z, 1)) .
I is su icien o p o e ha k≤˜κX(d). Choose an a bi a y α∈(α0,1) and
x, y ∈Xwi h kx−yk< d. Le y1be he elemen o he hal -line xy such ha
kx−y1k=d. The e o e he e exis s z∈Xwi h he p ope y ha kz−y1k ≤
α0dand B(x, 1) ∩B(y1, k)⊂B(z, 1).
I kx−yk ≤ αd we ha e B(x, 1)∩B(y, k)⊂B(x, 1) and xi sel is he equi ed
cen e .
I αd < kx−yk< d, hen ky−y1k<(1 −α)dand
B(x, 1) ∩B(y, k −(1 −α)d)⊂B(x, 1) ∩B(y1, k)⊂B(z, 1).
Mo eo e kz−yk ≤ (1 + α0−α)d.
Combining hese wo cases and bea ing P oposi ion 2.1 in mind we conclude
ha k−(1 −α)d≤˜κX(d). Taking αclose o 1 we comple e he p oo . £
F om P oposi ion 2.1 we also ob ain he ollowing p oposi ion.
P oposi ion 2.3. The unc ion ˜κXis nondec easing on [0,∞).
Nonexpansi i y ollows in a less i ial way.
P oposi ion 2.4. The unc ion ˜κXis nonexpansi e, i.e.
˜κX(d2)−˜κX(d1)≤d2−d1
o e e y 0≤d1≤d2.
P oo . Fix 0 < d1< d2and choose k < ˜κX(d2). The e o e he e exis s α∈(0,1)
such ha
∀x, y ∈X∃z∈X(kz−yk ≤ αd2∧B(x, 1) ∩B(y, k)⊂B(z, 1)) .
We will now p o e ha ˜κX(d1)≥k−(d2−d1).By P oposi ion 2.2 i is su icien
o conside x, y1∈Xwi h kx−y1k=d1. Le y2be he elemen on he hal -line
xy1such ha kx−y2k=d2. Hence
B(x, 1) ∩B(y1, k −(d2−d1)) ⊂B(x, 1) ∩B(y2, k)⊂B(z, 1)

250 RAFAEL ESP´
INOLA, ANDRZEJ WI´
SNICKI, AND JACEK WO´
SKO
o some z∈Xwi h kz−y2k ≤ αd2. We no ice ha y1=d2−d1
d2x+d1
d2y2and se
p=d2−d1
d2x+d1
d2z. The e o e kp−y1k=d1
d2kz−y2k ≤ αd1. Mo eo e i is no
di icul o see ha
B(x, 1) ∩B(y1, k −(d2−d1)) ⊂B(p, 1).
Thus ˜κX(d1)≥k−(d2−d1) and, aking sup emum o e k, we can deduce
˜κX(d1)≥˜κX(d2)−(d2−d1). £
We shall use P oposi ions 2.2, 2.3 and 2.4 in he nex sec ion.
3. The Op imali y Theo em
In his sec ion we show ha he e alua ion o G(A) gi en in Theo em 1.2 is
in a sense op imal.
Theo em 3.1. Le Xbe a no med space. Then o e e y d≥0and ε > 0 he e
exis se s A, G ⊂Xsuch ha
(3.1) |d (A,G)
(A)−d|< ε and G(A)
(A)<˜κX°d (A,G)
(A)±+ε.
P oo . Fix d≥0, ε > 0, and ake ksa is ying he inequali y ˜κX(d)< k <
˜κX(d) + ε
2and choose α∈(0,1) (no ice ha k > 1). Then he e exis x, y ∈X
wi h kx−yk=dsuch ha
(3.2) ∀z∈X(B(x, 1) ∩B(y, k)⊂B(z, 1) ⇒ kz−yk> αd).
We se A=B(x, 1) ∩B(y, k), G ={y}. I is clea ha G(A) = k, (A)≤1 and
d (A, G)≤d. We claim ha
(3.3) d (A, G)≥αd.
Indeed, i (A) = 1, hen x∈E0(A) and E0(A) is nonemp y. Using his ac and
(3.2), we can deduce d (A, G) = dis E0(A), y¡≥αd. I (A)<1, hen we se
0< δ < 1− (A) and, again, d (A, G)≥dis Eδ(A), y¡≥αd which p o es (3.3).
Mo eo e
(3.4) (A)≥1−(1 −α)d.
Assume con e sely, ha he e exis z∈Xand 0<1−(1 −α)dsuch ha
A⊂B(z, 0). In he case 0 ≤d≤1 we may use he ac ha kz−yk ≥ αd and
ha B(y, 1−d)⊆A. I we ake he poin u∈Ain he line zy which is he a hes
om zin B(y, 1−d), i can be seen ha ku−zk ≥ 1−d+αd = 1 −(1 −α)d,
which is a con adic ion wi h u∈A. In he case d > 1, le p∈Abe he elemen
o he hal -line xy such ha kx−pk= 1. Then ky−pk=d−1 and i ollows
RELATIVE CHEBYSHEV RADII AND HAUSDORFF MEASURES 251
om (3.2) ha kz−pk≥kz−yk−ky−pk>1−(1 −α)dwhich con adic s
p∈A. Thus (3.4) is p o ed.
Summa izing we ha e
(3.5) 1 −(1 −α)d≤ (A)≤1 and αd ≤d (A, G)≤d.
F om his ac and P oposi ions 2.3 and 2.4
G(A) = k≤˜κX(d) + ε
2≤˜κX°d (A,G)
α (A)±+ε
2≤
˜κX°d (A,G)
(A)±+d (A,G)
(A)²1
α−1³+ε
2.
Finally
G(A)
(A)≤˜κX(d) + ε
2
1−(1 −α)d≤
˜κX°d (A,G)
(A)±+d (A,G)
(A)1
α−1¡+ε
2
1−(1 −α)d
and om (3.5)
(3.6) αd ≤d (A, G)
(A)≤d
1−(1 −α)d.
Taking αsu icien ly close o 1 he o mula (3.1) holds. £
Theo em 1.2 also s a es
˜κX(d)≤in { G(A) : A, G ⊂X, (A) = 1 and d=d (A, G)}.
Using sligh ly mo e sub le a gumen s han in he p oo o Theo em 3.1 we may
imp o e he p e ious o mulae. We shall need he ollowing obse a ion, which is
easy o check.
Lemma 3.2. Le Ebe a con ex subse o a no med space X,y∈Xand
dis (y, E) = d > 0. Then o e e y ε > 0 he e exis y1, y2∈B(y, ε)such
ha dis (y1, E)≤max{d−ε
2,0}, dis (y2, E)≥d+ε
2.
Theo em 3.3. In any no med space X
˜κX(d) = in { G(A) : A, G ⊂X, (A) = 1 and d=d (A, G)}.
P oo . Fix d, ε > 0. As in he p oo o Theo em 3.1, o su icien ly la ge
α < 1, he e exis A⊂Xand y∈Xsuch ha ˜κX(d)≥ {y}(A)
(A)−εand αd ≤
d (A,{y})
(A)≤d
1−(1−α)d(see (3.6)). Mo eo e , i ollows om Lemma 3.3 ha we
can ix αclose o 1 o which he e exis y1, y2∈B(y, ε) wi h d (A, {y1})< d (A)
and d (A, {y2})> d (A). Since he unc ion x→d (A, {x}) is con inuous, he e
exis s z∈B(y, ε) wi h d (A, {z}) = d (A).Pu ¯
A=A
(A)and ¯z=z
(A). Then
252 RAFAEL ESP´
INOLA, ANDRZEJ WI´
SNICKI, AND JACEK WO´
SKO
d ¯
A, {¯z}¡=d, ¯
A¡= 1 and ˜κX(d)≥ {¯z}¯
A¡−2ε. This comple es he
p oo . £
No ice ha Theo em 3.3 gi es a geome ic desc ip ion o he Li schi z modulus.
4. The Case o Rela i e Hausdo Measu es o Noncompac ness
I is no di icul o see ha in any no med space X,
χG(A)≤χ(A) + lim
¯→0+d(H¯(A), G).
I Xis isome ic o C(K) o C0(Ω), hen by Theo em 1.3 ˜κX(d) = 1 + dand by
Theo em 1.2
χG(A) = χ(A) + lim
ε→0+d(Hε(A), G).
In his sec ion we ex end his esul o he space `1. Le us i s ecall he
no ion o minimal se s in oduced by Dom´ınguez Bena ides in [5].
De ini ion 4.1. Le Mbe a me ic space and ϕa measu e o noncompac ness.
A bounded, in ini e se A⊂Mis said o be minimal o he measu e ϕ(o , in
sho , ϕ-minimal) i ϕ(A) = ϕ(B) o e e y in ini e subse Bo A.
De ini ion 4.2. A measu e o noncompac ness ϕis said o be s ic ly minimaliz-
ing o a me ic space Mi o e e y bounded A⊂M he e exis s a ϕ- minimal
se B⊂Asuch ha ϕ(B) = ϕ(A).
I is known [3] ha he Hausdo measu e o noncompac ness χis s ic ly
minimalizing o a wide class o spaces including sepa able as well as e lexi e
Banach spaces. In pa icula , χis s ic ly minimalizing in `1space. Thus, o any
bounded se A⊂`1, he e exis s a χ-minimal sequence o A, ha is, a sequence
{xn}in Asuch ha χ(A) = χ({x1, x2, ...}).
We shall also need he ollowing esul conce ning `1space. We say ha a
bounded sequence {xn}o poin s o `1is coo dina e-wise con e gen o x∈`1i
o any i= 1,2, ..., limn→∞ xi
n=xi. Fo a ixed sequence {xn}in `1and a bi a y
y∈`1,we se
a(y, {xn}) = lim supn→∞ kxn−yk.
P oposi ion 4.3. ([9]). I {xn}is a bounded sequence in `1con e ging coo dina e-
wise o x∈`1, hen o any y∈`1
a(y, {xn}) = a(x, {xn}) + kx−yk.
To es ablish he main esul o his sec ion we shall use he ollowing lemmas.
RELATIVE CHEBYSHEV RADII AND HAUSDORFF MEASURES 253
Lemma 4.4. Fix x∈`1, > 0, ε ≥0and le {xn}be a sequence o poin s o
he ball B(x, ), which is coo dina e-wise con e gen o x0.I χ({x1, x2, x3, ...})≥
−ε, hen kx−x0k ≤ ε.
P oo . No ice ha a(x0,{xn})≥ −εand i ollows om P oposi ion 4.3 ha
kx−x0k= a(x, {xn})− a(x0,{xn})≤ −( −ε) = ε.
£
Lemma 4.5. Le Abe a bounded subse o a Banach space Xand ε > 0.
Then he e exis x1, ..., xn∈X, 1, ...., n>0such ha A⊂Sn
i=1 B(xi, i)
and χ(A∩B(xi, i)) ≥ i−ε, i = 1, ..., n.
P oo . Le A⊂Sm
i=1 B(yi, ai), whe e y1, ..., ym∈Xand a1, ..., am>0.I
χ(A∩B(yj, aj)) < aj−ε o some j∈ {1, ..., m}, he e exis z1, ..., zl∈X,
b1, ...., bl< aj−εsuch ha A∩B(yj, aj)⊂Sl
i=1 B(zi, bi).I , again,
χ(A∩B(yj, aj)∩B(zk, bk)) < bk−ε
o some k∈ {1, ..., l}we ha e A∩B(yj, aj)∩B(zk, bk)⊂Sp
i=1 B(ui, ci), whe e
u1, ..., up∈Xand c1, ..., cp< bk−ε<aj−2ε. A e a ini e numbe o s eps we
ob ain he desi ed co e . £
Lemma 4.6. Le x, y ∈`1,0< ≤kand ε > 0. The e o e he e exis s a ini e
se F⊂`1wi h he p ope ies B(x, )∩B(y, k)⊂B(F, )and F⊂B(y, k− +ε).
P oo . I ollows om Lemma 4.5 ha he e exis 1, ..., m∈`1and 1, ...., m≤
such ha B(x, )∩B(y, k)⊂Sm
i=1 B( i, i) and
χ(B(x, )∩B(y, k)∩B( i, i)) ≥ i−ε
2, i = 1, ..., m.
Fo each se Ui=B(x, )∩B(y, k)∩B( i, i) we selec a χ- minimal sequence
{xi
1, xi
2, ...} ⊂ Ui. We can assume, by aking subsequences i necessa y, ha he
sequences a e coo dina e wise con e gen o some w1, ..., wm∈`1, espec i ely.
Since χ{xi
1, xi
2, ...}¡≥ i−ε
2, i ollows om Lemma 4.4 ha k i−wik ≤ ε
2.
Mo eo e
kwi−yk= a(y, ¨xi
n©)− a(wi,¨xi
n©)≤k− i+ε
2
and hence
k i−yk≤k i−wik+kwi−yk ≤ k− i+ε, i = 1, ..., m.
Le us cons uc he se F={¯
1, ..., ¯
m}in he ollowing way: Fo i= 1, ..., m,
i k i−yk ≥ − i,¯
iis deno ed as he poin o he segmen [ i, y] sa is ying he
equali y k i−¯
ik= − i, − i≥0. I k i−yk< − iwe se ¯
i=y. I is no