Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 139597, 9 pages
doi:10.1155/2011/139597
Research Article
Convexities and Existence of the Farthest Point
Z. H. Zhang and C. Y. Liu
College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201600, China
Correspondence should be addressed to Z. H. Zhang, zhzmath@gmail.com
Received 10 July 2011; Revised 20 September 2011; Accepted 20 September 2011
Academic Editor: Toka Diagana
Copyright q 2011 Z. H. Zhang and C. Y. Liu. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Five counterexamples are given, which show relations among the new convexities and some
important convexities in Banach space. Under the assumption that Banach space X is nearly very
convex, we give a sufficient condition that bounded, weakly closed subset of X has the farthest
points. We also give a sufficient condition that the farthest point map is single valued in a residual
subset of X when X is very convex.
1. Introduction
Let X be a Banach space, and let X ∗ be its dual space. Let us denote by BX and SX the
closed unit ball and the unit sphere of X, respectively. Let x ∈ SX, Σx {x∗ ∈ SX ∗ :
m}. Let B be a
x∗ x 1}. For any sequence {xn } ⊂ X, define sepxn ≡ inf{xn − xm : n /
bounded subset of X. We define a real-valued function rB : X → R by
rB x sup x − y : y ∈ B
1.1
and call rB x the farthest distance from x to B. The function rB is convex and Lipschitzcontinuous. In fact, |rB x − rB y| ≤ x − y for all x, y ∈ X.
A point z ∈ B is called a farthest point of B if there exists an x ∈ X such that x − z rB x.
The mapping FB : X → 2B defined by FB x {z ∈ B : x − z rB x} is called the
farthest point map of B.
The existence of a farthest point of B is equivalent to the fact that the set
D {x ∈ X : x − z rB x for some z ∈ B}
1.2
2
Abstract and Applied Analysis
is nonempty. In 1–5, the existence of a farthest point of B is studied. Edelstein 2 showed
that if X is uniformly convex space, then the set D defined above is dense in X. Asplund 1
showed that if X is both reflexive and locally uniformly rotund LUR, then the set D is dense
in X. Lau 4 proved that if B is weakly compact subset of X, then D contains a dense Gδ set
of X, and if X is reflexive Banach space, then for every bounded, weakly closed subset B in
X, D contains a dense Gδ subset of X.
We say that X is strongly convex resp., very convex/nearly strongly convex/nearly
very convex if any x ∈ SX and {xn } ⊂ BX with x∗ xn → 1 as n → ∞ for some x∗ ∈ Σx
w
imply xn → x as n → ∞ resp., xn −→ x as n → ∞/{xn } is relatively compact/{xn } is
weakly relatively compact.
The author 6 proved that strong convexity resp., very convex/nearly strong
convexity/nearly very convex has important applications in approximation theory.
Bandyopadhyay et al. 7 also proposed two generalizations of LUR and weakly locally
uniform rotundity WLUR, which were called almost locally uniform rotundity ALUR
and weakly almost locally uniform rotundity WALUR. A Banach space X is said to be
∗
} ⊂ BX ∗ , the condition
ALUR resp., WALUR if for any x ∈ SX, {xn } ⊂ BX and {xm
w
∗
limm limn xm
xn x/2 1 implies xn → x reps., xn −→ x. Recently, we have proved
that ALUR and strong convexity, WALUR and very convex are equivalent, respectively 8.
Sullivan 9 defined very rotund space. A Banach space X is known as very rotund if no
x∗ ∈ SX ∗ is simultaneously a norming element for some x ∈ SX and x∗∗ ∈ SX ∗∗ , where
x/
x∗∗ . The author 10 proved that very rotund space coincides with very convex space. By
6–12, we know that the four new convexities mentioned above have a lot of good properties
and applications.
It is known that LUR, WLUR, midpoint locally uniform rotundity MLUR and
weakly midpoint locally uniform rotundity WMLUR are four important convexities in the
geometric theory of Banach spaces. By 9–12, the relation of the convexities mentioned above
is shown in Figure 1 below.
The structure of this paper is as follows. In Section 2, We will give five counterexamples, which show the relations among the four new convexities mentioned above and LUR,
WLUR, MLUR, WMLUR, and rotund R.
In Section 3, we prove that if X is nearly very convex space and if for every x ∈ X, there
exists x0∗ ∈ ∂rB x which attains its norm, then for every bounded, weakly closed subset B of
X, the set D defined above contains a Gδ subset of X, which improves the results of Edelstain
2, Asplund 1, and Lau 4. Finally, we also prove a sufficient condition that the farthest
point map FB is single valued in a residual subset of X when X is very convex.
2. Some Counterexamples about Convexities
Lemma 2.1 see 13. Let {xα : α ∈ D} be a net in X if for every ε > 0, there exists αs ∈ D such
that the tail {xα : α ≥ αs } has a finite ε-net, then {xα } is a relatively compact subset in X.
Lemma 2.2. If X is nearly strongly convex space, then X has Kadec property, that is, if whenever
w
x ∈ SX and {xα } is a net in SX such that xα −→ x, then xα → x. Particularly, X has property
w
H, that is, if whenever x ∈ SX and {xn } is a sequence in SX such that xn −→ x, then xn → x.
w
Proof. Suppose that net {xα : α ∈ D} ⊂ SX, x ∈ SX such that xα −→ x, we will prove that
xα → x.
Abstract and Applied Analysis
LUR
3
Strong convexity
MLUR
Nearly strong convexity
R
Nearly very convex
WLUR
WMLUR
Very convex
Figure 1: The relationship between the convexities.
Case 1. If for every ε > 0, there exists αs ∈ D such that {xα : α ≥ αs } has a finite ε-net, by
w
Lemma 2.1 and xα −→ x, we may obtain that xα → x.
Case 2. If for every ε0 > 0 such that all tails of net {xα : α ∈ D} have no finite ε0 -net, we take
w
f ∈ SX ∗ with f in Σx. Since xα −→ x, fxα → fx 1. Choosing {βn } ⊂ D, β1 ≤ β2 ≤
· · · ≤ βn ≤ · · · , we know that
fxα > 1 −
1
,
n
2.1
for any α ∈ D with α ≥ βn . Take α1 ∈ D such that α1 ≥ β1 . For xα1 and β2 , we can choose
α2 ∈ D, α2 ≥ β2 such that xα2 − xα1 ≥ ε0 . Otherwise, {xα : α ≥ β2 } ⊂ {x ∈ X : x − xα1 < ε0 },
that is, the tail {xα : α ≥ β2 } has finite ε0 -net. This is a contradiction with the assumption. For
xα1 , xα2 , and β3 , there exists α3 ∈ D, α3 ≥ β3 such that xα3 − xα1 ≥ ε0 and xα3 − xα2 ≥ ε0 .
Otherwise, {xα : α ≥ β3 } ⊂ {x ∈ X : x − xα1 < ε0 } ∪ {x ∈ X : x − xα2 < ε0 }, that
is, the tail {xα : α ≥ β3 } has finite ε0 -net. This is a contradiction with the assumption.
According to the same method, we may choose a sequence {xαn } ⊂ {xα : α ∈ D} such
that
sepxαn inf{xαn − xαm : n /
m} ≥ ε0 ,
αn ≥ βn , n 1, 2, . . . .
2.2
Hence, we know that {xαn } is not relatively compact.
On the other hand, by 2.1, we have that
1 ≥ fxαn > 1 −
1
.
n
2.3
This shows that fxαn → 1. Since X is nearly strongly convex, {xαn } is relatively compact
which is a contradiction.
Example 2.3. There exists an MLUR space which is not a nearly strongly convex space.
4
Abstract and Applied Analysis
Recall the equivalent norm defined on c0 by Smith 14. For k ∈ N, define a mapping
Vk : c0 → R1 , Vk x Vk ξ1 , ξ2 , . . . sup{|ξ1 − ξi | : i ≥ k}. Let {αn } be a sequence of positive
2
real numbers, and ∞
n2 αn 1. Define two mappings V and T from c0 → l2 as follows:
V x α2 V2 x, α3 V3 x, . . .,
T x ξ1 , α2 ξ2 , α3 ξ3 , . . ..
Since {Vk x} and {ξk } are both bounded sequences, and
{Vk x} and T x are in l2 . For x in c0 , let
∞
n2
2.4
α2n 1, then we know that
1/2
.
xG x2∞ V x22 T x22
2.5
Since T is one to one continuous,
√ linear, and V x y ≤ V x V y for all x, y in c0 , the · G
is a norm, and · ∞ ≤ · G ≤ 6 · ∞ . This shows that · G is an equivalent norm on c0 .
Smith shows that c0 , · G is MLUR. We say that c0 , · G has no property H.
√
w
Indeed, let x e1 , xn e1 en , then xn G → xG 3 and xn −→ x, but xn x. By
Lemma 2.2, we know that c0 , · G is not nearly strongly convex.
Example 2.4. There exists a very convex space which is not nearly strongly convex space.
Recall the equivalent norm on Hilbert space by Troyanski in Isratescu 15. Let X be
a Hilbert space, and let {ei }∞
i0 be an orthogonal basis. For any x ∈ X, x λ0 e0 λ1 e1 · · · λn en · · · , let
x21 max |λ0 |2 |λ2 |2 · · · |λ2n |2 · · · , |λ1 |2 |λ3 |2 · · · |λ2n1 |2 · · · .
2.6
It is obvious that · 1 is an equivalent norm on X. Further, we set
x x21
∞
1
i1
2i
1/2
|λi |
2
.
2.7
Clearly, this is again a norm on X which is an equivalent original one. Troyanski shows that
X, · is R and reflexive 15, but it has no property H. Hence, X, · is very convex, but
is not nearly strongly convex by Lemma 2.2.
Example 2.5. There exists a nearly very convex space which is neither very convex space nor
nearly strongly convex space.
For any x ∈ l2 , let
1
|x| max √ x2 , x∞ .
2
2.8
√
Since 1/ 2x2 ≤ |x| ≤ x2 , |x| is an equivalent norm on l2 . Since l2 , | · | is reflexive,
we know that l2 , | · | is nearly very convex space. We say that l2 , | · | has no property H.
Indeed, let x e1 , xn e1 en , then |x| |xn | 1, but |xn − xm | 1. Hence, B l2 , | · |
is not nearly strongly convex by Lemma 2.2. We say that l2 , | · | is not very convex. In fact,
Abstract and Applied Analysis
5
let e1,2 1, 1, 0, . . ., e1,3 1, 0, 1, 0, . . ., then |e1,2 e1,3 | 2. This shows that l2 , | · | is
not R, and therefore, l2 , | · | is not very convex.
Example 2.6. There exists a strongly convex space which is not WLUR space.
Let E l2 , · , where x a1 , a2 , . . . ∈ l2 ,
2
1/2 2
a 2
an
2
2
2
··· ··· .
x |a1 | a2 a3 · · ·
2
n
2
2.9
Let X Σ ⊕ El2 . In 16, it is proved that X is 2R, but is not KUR. Since 2R implies R, reflexive
and property H, we get that X is strongly convex space.
∞
X is not WLUR space. Indeed, let {ei }√
1 , e1 , 0, . . ., xn i1 be the natural basis, and x e√
1/2e1 en , e1 , 0, . . ., then limn → ∞ xn 2 x and limn → ∞ xn x 8. However,
1
d xn , span{x} inf xn − y : y ∈ span{x} ≥ .
2
2.10
Choose f ∈ SX ∗ such that fxn ≥ 1/2, fy 0, for all y ∈ span{x}. It is easily proved
that
x
xn √1 lim x xn 2,
lim
n → ∞ x
xn 2n→∞
2.11
√
but fx/x xn /xn ≥ |fxn /xn | 1/2 2. This shows that X is not WLUR space.
Example 2.7. There exists a nearly strongly convex space which is not a strongly convex space.
For x x1 , x2 in R2 , let x |x1 | |x2 |, then R2 , · is reflexive and has property H.
Hence, it is nearly strongly convex. It is easy to prove that R2 , · is not R space. Therefore
R2 , · is not strongly convex.
Remark 2.8 Smith 17. gave three examples l2 , · W , l2 , · A , l1 , · H . He shows that
l2 , · W is WLUR not MLUR, and l2 , · A is MLUR not WLUR, and l1 , · H is R not
MLUR. It is easily proved that l1 , · H is not WMLUR either.
By the above five counterexamples and Remark 2.8, we know that, except for very
convex implied WMLUR, none of the above converse implied relations in the diagram is
generally true.
3. Convexities and Existence of the Farthest Point
Before proceeding to this part, let’s recall that the subdifferential of convex function f on
Banach space X is defined by
∂fx x∗ ∈ X ∗ : x∗ y − x ≤ f y − fx, ∀y ∈ X .
x → ∂fx is called subdifferential mapping.
3.1
6
Abstract and Applied Analysis
Remark 3.1. It was shown in 4 that if B is a bounded closed subset in Banach space X, then
for any x ∈ X and x∗ ∈ ∂rB x, we have that x∗ ≤ 1 and thus
sup{x∗ x − z : z ∈ B} ≤ x∗ · rB x ≤ rB x.
3.2
Hence, for any x ∈ X and x∗ ∈ ∂rB x, we have that
inf x∗ z − x ≥ −rB x.
z∈B
3.3
Lemma 3.2 Lau 4. Let X be a Banach space and B a bounded subset in X, then the set
F
x ∈ X : inf x∗ z − x > −rB x for some x∗ ∈ ∂rB x
z∈B
3.4
is a first category in X.
Theorem 3.3. Let X be a nearly very convex Banach space and B a bounded, weakly closed subset of
X. Further, for any x ∈ X, if there exists an x0∗ ∈ ∂rB x which attains its norm, then
D {x ∈ X : x − z rB x for some z ∈ B}
3.5
contains a dense Gδ set of X. In particular, the set of farthest points of B is nonempty.
Proof. Define F as in Lemma 3.2 and
Fn 1
x ∈ B : inf x∗ z − x ≥ −rB x for some x∗ ∈ ∂rB x .
z∈B
n
3.6
Let Q X \ F, then
QX\
∞
Fn n1
∞
X \ Fn ,
3.7
n1
where each X \ Fn is an open, dense subset in B. Hence, Q is a dense Gδ set in X.
Now, we prove Q ⊂ D as follows. For any x ∈ Q, take x0∗ ∈ ∂rB x such that x0∗ attains
its norm. Since infz∈B x0∗ z − x ≥ −rB x, by the definition of Q, we have that infz∈B x0∗ z − x ≤
−rB x. Take sequence {zn } ⊂ B such that
lim x∗ zn
n→∞ 0
− x −rB x.
3.8
Clearly, zn − x/−rB x ∈ BX. Given that X is nearly very convex and that B is weakly
w
closed, there exist z0 ∈ B and subsequence {znk } ⊂ {zn } such that znk − x/−rB x −→
Abstract and Applied Analysis
7
w
z0 − x/−rB x as k → ∞, that is, znk −→ z0 as k → ∞. Hence, x0∗ z0 − x −rB x. It
follows that
rB x ≥ x − z0 ≥ x0∗ z0 − x rB x.
3.9
Thus, z0 − x rB x. This shows that Q ⊂ D.
Corollary 3.4 Lau 4. If X is a reflexive Banach space. Then for every bounded, weakly closed
subset B in X, the set
D {x ∈ X : x − z rB x for some z ∈ B}
3.10
contains a dense Gδ set of X, and hence, the set of farthest points of B is nonempty.
Corollary 3.5 Asplund 1. If X is a reflexive LUR Banach space, then Corollary 3.4 holds for
every bounded closed subset B in X.
Theorem 3.6. Let X be a very convex Banach space and B a bounded, weakly closed subset of X. For
any x ∈ X, if there exists an x0∗ ∈ ∂rB x which attains its norm, then the farthest point map FB is
single valued in a residual subset of X.
Proof. By Theorem 3.3,
Q X\F ∗
∗
x ∈ X : sup x x − z rB x for any x ∈ ∂rB x
3.11
z∈B
is a dense Gδ subset of X, where F is defined as in Lemma 3.2.
Now we prove that FB x is single valued for all x ∈ Q.
z2 such
If FB is not single valued on Q, then there are x0 ∈ Q and z1 , z2 ∈ B with z1 /
that x0 − zi rB x0 , i 1, 2. By Hahn-Banach theorem, we have xi∗ ∈ SX ∗ , i 1, 2 such
that
xi∗ x0 − zi x0 − zi ,
i 1, 2.
3.12
For any z ∈ B, y ∈ X, we have that
xi∗ y − x0 xi∗ y − z − xi∗ x0 − z
≤ sup sup x∗ y − z − xi∗ x0 − z
z∈B x∗ ∈BX ∗ 3.13
rB y − xi∗ x0 − z .
Thus,
sup xi∗ x0 − z ≤ rB y − xi∗ y − x0 .
z∈B
3.14
8
Abstract and Applied Analysis
Since x0 ∈ Q, rB x0 ≤ rB y − xi∗ y − x0 . This shows that xi∗ ∈ ∂rB x0 . Let x0∗ 1/2x1∗ 1/2x2∗ , then x0∗ ∈ ∂rB x0 due to the convexity of ∂rB x0 , and supz∈B x0∗ x0 − z rB x0 .
Take a sequence {zn } ⊂ B such that
lim x∗ x0
n→∞ 0
− zn rB x0 .
3.15
It follows that limn → ∞ xi∗ x0 − zn rB x0 , i 1, 2. By 3.12, xi∗ ∈ Σx0 − zi /rB x0 , i w
1, 2. Because X is very convex, x0 − zn /rB x0 −→ x0 − zi /rB x0 , i 1, 2 as n → ∞.
According to uniqueness of weak limit point, we have that z1 z2 , which is a contradiction.
Remark 3.7. For closed-convex subset B and bounded closed, relatively weakly compact K
in X, Ni and Li 18 proved that the set of all points in B such that the farthest problem
max{x, K} is well posed is a dense Gδ subset in B provided that B is both strictly convex
and Kadec with respect to K. This shows that the farthest point map FB is single valued
in a residual subset of X. By Example 2.4 in this paper, there exists a Banach space where
assumptions in Theorem 3.6 are satisfied, but its unit ball BX is not Kadec. Let BX B K, then we know that conditions of Theorem 3.6 are different from conditions of the result by
Ni and Li. Hence, the result by Ni and Li does not imply Theorem 3.6.
Acknowledgment
The authors would like to extend their heartfelt gratitude to those dear referees for their
valuable suggestions.
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