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Abstract and Applied Analysis
Volume 2011, Article ID 139597, 9 pages
http://dx.doi.org/10.1155/2011/139597
Research Article

Convexities and Existence of the Farthest Point

College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201600, China

Received 10 July 2011; Revised 20 September 2011; Accepted 20 September 2011

Academic Editor: Toka Diagana

Copyright © 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.

Abstract

Five counterexamples are given, which show relations among the new convexities and some important convexities in Banach space. Under the assumption that Banach space is nearly very convex, we give a sufficient condition that bounded, weakly closed subset of has the farthest points. We also give a sufficient condition that the farthest point map is single valued in a residual subset of when is very convex.

1. Introduction

Let be a Banach space, and let be its dual space. Let us denote by and the closed unit ball and the unit sphere of , respectively. Let . For any sequence , define . Let be a bounded subset of . We define a real-valued function by and call the farthest distance from to . The function is convex and Lipschitz-continuous. In fact, for all .

A point is called a farthest point of if there exists an such that .

The mapping defined by is called the farthest point map of .

The existence of a farthest point of is equivalent to the fact that the set is nonempty. In [15], the existence of a farthest point of is studied. Edelstein [2] showed that if is uniformly convex space, then the set defined above is dense in . Asplund [1] showed that if is both reflexive and locally uniformly rotund (LUR), then the set is dense in . Lau [4] proved that if is weakly compact subset of , then contains a dense set of , and if is reflexive Banach space, then for every bounded, weakly closed subset in , contains a dense subset of .

We say that is strongly convex (resp., very convex/nearly strongly convex/nearly very convex) if any and with as for some imply as (resp., as / is relatively compact/ 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 is said to be ALUR (resp., WALUR) if for any and , the condition implies (reps., . 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 is known as very rotund if no is simultaneously a norming element for some and , where . The author [10] proved that very rotund space coincides with very convex space. By [612], 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 [912], the relation of the convexities mentioned above is shown in Figure 1 below.

139597.fig.001
Figure 1: The relationship between the convexities.

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 is nearly very convex space and if for every , there exists which attains its norm, then for every bounded, weakly closed subset of , the set defined above contains a subset of , which improves the results of Edelstain [2], Asplund [1], and Lau [4]. Finally, we also prove a sufficient condition that the farthest point map is single valued in a residual subset of when is very convex.

2. Some Counterexamples about Convexities

Lemma 2.1 (see [13]). Let be a net in if for every , there exists such that the tail has a finite -net, then is a relatively compact subset in .

Lemma 2.2. If is nearly strongly convex space, then has Kadec property, that is, if whenever and is a net in such that , then . Particularly, has property , that is, if whenever and is a sequence in such that , then .

Proof. Suppose that net such that , we will prove that .Case 1. If for every , there exists such that has a finite -net, by Lemma 2.1 and , we may obtain that .Case 2. If for every such that all tails of net have no finite -net, we take with in . Since . Choosing , we know that for any with . Take such that . For and , we can choose such that . Otherwise, , that is, the tail has finite -net. This is a contradiction with the assumption. For , and , there exists such that and . Otherwise, , that is, the tail has finite -net. This is a contradiction with the assumption. According to the same method, we may choose a sequence such that Hence, we know that is not relatively compact.
On the other hand, by (2.1), we have that This shows that . Since is nearly strongly convex, is relatively compact which is a contradiction.

Example 2.3. There exists an MLUR space which is not a nearly strongly convex space.
Recall the equivalent norm defined on by Smith [14]. For , define a mapping . Let be a sequence of positive real numbers, and . Define two mappings and from as follows: Since and are both bounded sequences, and , then we know that and are in . For in , let Since is one to one continuous, linear, and for all in , the is a norm, and . This shows that is an equivalent norm on .
Smith shows that is MLUR. We say that has no property H. Indeed, let , then and , but . By Lemma 2.2, we know that 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 be a Hilbert space, and let be an orthogonal basis. For any , let It is obvious that is an equivalent norm on . Further, we set Clearly, this is again a norm on which is an equivalent original one. Troyanski shows that is R and reflexive [15], but it has no property H. Hence, 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 , let Since ( is an equivalent norm on . Since is reflexive, we know that is nearly very convex space. We say that has no property H. Indeed, let , then , but . Hence, B is not nearly strongly convex by Lemma 2.2. We say that is not very convex. In fact, let , then . This shows that is not R, and therefore, is not very convex.

Example 2.6. There exists a strongly convex space which is not WLUR space.
Let , where , Let . In [16], it is proved that is 2R, but is not KUR. Since 2R implies R, reflexive and property H, we get that is strongly convex space.
is not WLUR space. Indeed, let be the natural basis, and , then and . However, Choose such that . It is easily proved that but . This shows that is not WLUR space.

Example 2.7. There exists a nearly strongly convex space which is not a strongly convex space.
For in , let , then is reflexive and has property H. Hence, it is nearly strongly convex. It is easy to prove that is not R space. Therefore is not strongly convex.

Remark 2.8 (Smith [17]). gave three examples . He shows that is WLUR not MLUR, and is MLUR not WLUR, and is R not MLUR. It is easily proved that 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 on Banach space is defined by

is called subdifferential mapping.

Remark 3.1. It was shown in [4] that if is a bounded closed subset in Banach space , then for any and , we have that and thus Hence, for any and , we have that

Lemma 3.2 (Lau [4]). Let be a Banach space and a bounded subset in , then the set is a first category in .

Theorem 3.3. Let be a nearly very convex Banach space and a bounded, weakly closed subset of . Further, for any , if there exists an which attains its norm, then contains a dense set of . In particular, the set of farthest points of is nonempty.

Proof. Define as in Lemma 3.2 and Let , then where each is an open, dense subset in . Hence, is a dense set in .
Now, we prove as follows. For any , take such that attains its norm. Since , by the definition of , we have that . Take sequence such that Clearly, . Given that is nearly very convex and that is weakly closed, there exist and subsequence such that as , that is, as . Hence, . It follows that Thus, . This shows that .

Corollary 3.4 (Lau [4]). If is a reflexive Banach space. Then for every bounded, weakly closed subset in , the set contains a dense set of , and hence, the set of farthest points of is nonempty.

Corollary 3.5 (Asplund [1]). If is a reflexive LUR Banach space, then Corollary 3.4 holds for every bounded closed subset in .

Theorem 3.6. Let be a very convex Banach space and a bounded, weakly closed subset of . For any , if there exists an which attains its norm, then the farthest point map is single valued in a residual subset of .

Proof. By Theorem 3.3, is a dense subset of , where is defined as in Lemma 3.2.
Now we prove that is single valued for all .
If is not single valued on , then there are and with such that . By Hahn-Banach theorem, we have such that For any , we have that Thus, Since . This shows that . Let , then due to the convexity of , and . Take a sequence such that It follows that . By (3.12), . Because is very convex, ( as . According to uniqueness of weak limit point, we have that , which is a contradiction.

Remark 3.7. For closed-convex subset and bounded closed, relatively weakly compact in , Ni and Li [18] proved that the set of all points in such that the farthest problem is well posed is a dense subset in provided that is both strictly convex and Kadec with respect to . This shows that the farthest point map is single valued in a residual subset of . By Example 2.4 in this paper, there exists a Banach space where assumptions in Theorem 3.6 are satisfied, but its unit ball is not Kadec. Let , 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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