2012
DOI: 10.1155/2012/724120
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Convexity and Proximinality in Banach Space

Abstract: By the continuity of preduality map, we give some necessary and sufficient conditions of the strongly convex and very convex spaces, respectively. Using nearly strong convexity of X, we give some equivalent conditions that every element in X is strongly unique of order p, bounded strongly unique of order p, and locally strongly unique of order p.

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Cited by 6 publications
(7 citation statements)
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“…They have been studied, e.g., in [1,7,16,18,19,23]. In particular, [7,19,20,22] contain applications to approximation theory.…”
Section: Definitions Notation and An Earlier Resultsmentioning
confidence: 99%
See 1 more Smart Citation
“…They have been studied, e.g., in [1,7,16,18,19,23]. In particular, [7,19,20,22] contain applications to approximation theory.…”
Section: Definitions Notation and An Earlier Resultsmentioning
confidence: 99%
“…This multivalued mapping, denoted by D −1 for obvious reasons, sends f ∈ S 0 (X * ) (where S 0 (X * ) denotes the subset of S X * consisting of all functionals that attain their supremum on B X ) to the set {x ∈ S X : f (x) = 1}. Predecessors of the results here, formulated in terms of the duality mapping are, e.g., in [5], and of the pre-duality mapping D −1 : S 0 (X * ) → S X , in [20].…”
mentioning
confidence: 99%
“…Example 2. 13 shows that the answer to [3, Question 3.9] is negative. Furthermore, Theorem 2.6(1) provides the proper condition for which the conclusion to [3, Question 3.9] holds.…”
Section: Theorems and Examplesmentioning
confidence: 99%
“…The τ -strongly Chebyshev, approximative τ -compactness, τ -strong proximinality, and proximinality in Banach space are basic and important properties of approximation theory, and the topic of much research [2, 4-7, 11, 14], [3,8], [13], where τ is the norm topology or the weak topology. Therefore, it is critical to clarify the relationship between these proximinalities.…”
Section: Introductionmentioning
confidence: 99%
“…Then we have LUR ⇒ strongly convex ⇒ nearly strongly convex ⇒ nearly very convex, and these four concepts are different (see [ZL11, Examples 2.5, 2.6, and 2.7]) (for examples outside the context of reflexive spaces -all of them nearly very smoothconsider [Dr14, Theorem 1], where it is proved that every infinite-dimensional Banach space with separable dual admits an equivalent wLUR norm which is not LUR: It is obvious that every wLUR space is nearly very convex; this wLUR equivalent norm cannot be nearly strongly convex, since this last property implies property (H) that, together with wLUR, implies LUR, see below). The concepts nearly strongly convex and nearly very convex are discussed, e.g., in [BLLN08], [FW01], [GM11], [ZL11], [ZL12], [ZMLG15], and [ZS09], and they are related to questions of approximation in Banach spaces. We may mention, for example, a characterization of nearly strict convexity in terms of the preduality mapping:…”
Section: Introductionmentioning
confidence: 99%