On P- and p-Convexity of Banach Spaces

and Applied Analysis 3 Lemma 2.7 Goebel-Kirk . Let X be a Banach space. For each ε ∈ ε0 X , 2 , one has the equality δX 2 − 2δX ε 1 − ε/2. Lemma 2.8 Ullán . Let X be a Banach space. For each 0 ≤ ε2 ≤ ε1 < 2 the following inequality holds: δX ε1 − δX ε2 ≤ ε1 − ε2 / 2 − ε1 . Using these lemmas we obtain: Theorem 2.9. Let X be a Banach space which satisfies δX 1 > 0, that is, ε0 X < 1. Then X is P(3)-convex. Moreover, there exists a Banach space X with ε0 X 1 which is not P(3)-convex. Proof. Let t0 2 − √ 2 − ε0 X . Clearly ε0 X < t0 < 1. Let x, y, z ∈ SX , and suppose that ‖x − y‖ > 2 − 2δX t0 and ‖x − z‖ > 2 − 2δX t0 . By Lemma 2.7, we have ∥ ∥ ∥ x y 2 ∥ ∥ ∥ ≤ 1 − δ 2 − 2δX t0 1 − ( 1 − t0 2 ) t0 2 . 2.6 Similarly ‖ x z /2‖ ≤ t0/2. Hence we get ∥z − y∥ ≤ ‖z x‖ ∥x y∥ ≤ 2t0. 2.7 Finally, from Lemma 2.8 it follows that δX t0 δX t0 − δX ε0 X ≤ t0 − ε0 X 2 − t0 √ 2 − ε0 X − 1 1 − t0. 2.8 Then ‖y − z‖ ≤ 2t0 ≤ 2 − 2δX t0 , and thus X is P 3 -convex. Now consider for each 1 < p < ∞ the space lp,∞ defined as follows. Each element x {xi}i ∈ lp may be represented as x x − x−, where the respective ith components of x and x− are given by x i max{xi, 0} and x− i max{−xi, 0}. Set ‖x‖p,∞ max{‖x ‖p, ‖x‖p} where ‖ · ‖p stands for the lp-norm. The space lp,∞ lp, ‖ · ‖p,∞ satisfies ε0 lp,∞ 1 see 3 . On the other hand let x1 e1 − e3, x2 −e1 e2, x3 −e2 e3 ∈ Slp,∞ , where {ei}i is the canonical basis in lp. These points satisfy that ‖xi − xj‖p,∞ 2, i / j. Thus lp,∞ is not P 3.2 -convex. It is known that if a Banach space X satisfies ε0 X < 1, then X has normal structure as well as P 3 -convexity. The space X lp,∞ is an example of a Banach space with ε0 X 1 which does not have normal structure see 3 and is not P 3 -convex. Kottman also proved in 1 that every uniformly smooth space is a P-convex space. We obtain a generalization of this fact. Before we show this result we recall the next concept. Definition 2.10. The modulus of smoothness of a Banach spaceX is the function ρX : 0,∞ → 0,∞ defined by ρX t sup { 1 2 ∥x ty ∥ ∥x − ty∥ − 2 : x, y ∈ SX } 2.9 for each t ≥ 0. X is called uniformly smooth if limt→ 0 ρX t /t 0. 4 Abstract and Applied Analysis The proofs of the following lemmas can be found in 4, 5 . Lemma 2.11. For every Banach space X, one has limt→ 0ρX t /t 1/2 ε0 X∗ . Lemma 2.12. Let X be a Banach space. X is P(3)-convex if and only if X∗ is P(3)-convex. By Theorem 2.9 and by the previous lemmas we deduce the next result. Corollary 2.13. If X is a Banach space satisfying limt→ 0ρX t /t < 1/2, then X is P(3)-convex. With respect to P 4 -convex spaces we have this result, which is easy to prove. Proposition 2.14. If X is a Banach space P ε, 4 -convex, then ε0 X ≤ 2 − ε, and hence X is uniformly nonsquare. In fact, in bidimensional normed spaces, P 4 -convexity and uniform nonsquareness coincide. The proof of this involves many calculations and can be seen in 6 . Another technical proof see 6 shows that if X is a bidimensional normed space, thenX is always P 1,5 -convex. Hence the spaceX R2, ‖ · ‖∞ is P 1,5 -convex and ε0 X 2, and thus P 5 -convexity does not imply uniform squareness. 2.3. Relation between U-Spaces and P-Convex Spaces In this section we show that P-convexity follows from U-convexity. The following concept was introduced by Lau in 1978 7 . Definition 2.15. A Banach space X is called a U-space if for any ε > 0 there exists δ > 0 such that x, y ∈ SX, f ( x − y > ε, for some f ∈ ∇ x ⇒ ∥∥∥ x y 2 ∥∥∥ ≤ 1 − δ, 2.10 where for each x ∈ X ∇ x f ∈ SX∗ : f x ‖x‖ } . 2.11 The modulus of this type of convexity was introduced by Gao in 8 and further studied by Mazcuñán-Navarro 9 and Saejung 10 . The following result is proved in 8 . Lemma 2.16. Let X be a Banach space. If X is U-space, then X is uniformly nonsquare, From the above we obtain the next theorem which is a generalization of Kottman’s result, who showed in 1 that P 3 -convexity follows from uniform convexity. Theorem 2.17. If X is a U-space, then X is P( 3)-convex. Proof. By Lemma 2.16 we have that there exists α > 0 such that for all ξ, η ∈ SX min ∥ξ − η∥,∥ξ η∥ ≤ 2 − α. 2.12 Abstract and Applied Analysis 5and Applied Analysis 5 Since X is a U-space, for ε α/2 there exists δ > 0 such that x, y ∈ SX, f ( x − y ≥ α 2 , for some f ∈ ∇ x ⇒ ∥ ∥ ∥∥ x y 2 ∥ ∥ ∥∥ ≤ 1 − δ. 2.13 We claim thatX is P β, 3 -convex, where β min{α, δ}. Indeed, proceeding by contradiction, assume that there exist x, y, z ∈ SX such that min ∥x − y∥, ‖x − z‖,∥y − z∥ > 2 − β. 2.14 Define w −y and u −z, and let f ∈ ∇ w . If f w − x ≥ α/2, then ∥ ∥ ∥ w x 2 ∥ ∥ ∥ < 1 − δ. 2.15 Therefore 2 − δ ≤ 2 − β < ‖x − y‖ < 2 − 2δ, which is not possible. Hence f w − x < α/2. Similarly we prove f w u < α/2. Also ‖x u‖ ‖x−z‖ > 2−β ≥ 2−α, and hence, by 2.12 we have f x − u ≤ ‖x − u‖ ≤ 2 − α. By the above we have 2 2f w f w − x f x − u f u w < α 2 2 − α α 2 2 2.16 which is a contradiction. 2.4. The Dual Concept of P-Convexity In 1 , Kottman introduces a property which turns out to be the dual concept of P-convexity. In this section we characterize the dual of a P-convex space in an easier way. We begin by showing Kottman’s characterization. Definition 2.18. Let X be a Banach space and ε > 0. A convex subset A of BX is said to be ε-flat if A ⋂ 1 − ε BX ∅. A collection D of ε-flats is called complemented if for each pair of ε-flats A and B in D we have that A ⋃ B has a pair of antipodal points. For any n ∈ N we define F n,X inf { ε > 0 : BX has a complemented collection D of ε-flats such that Card D n } . 2.17 Theorem 2.19 Kottman . Let X be a Banach space and n ∈ N. Then a F n,X∗ 0 ⇔ P n,X 1/2. b P n,X∗ 1/2 ⇔ F n,X 0. Now we define P-smoothness and prove that it turns out to be the dual concept of P-convexity. The advantage of this characterization is that it uses only simple concepts, and one does not need ε-flats. Besides in the proof of the duality we do not need Helly’s theorem nor the theorem of Hahn-Banach, as Kottman does in Theorem 2.19. 6 Abstract and Applied Analysis Definition 2.20. Let X be a Banach space and δ > 0. For each f, g ∈ X∗ set S f, g, δ {x ∈ BX : f x ≥ 1 − δ, g x ≥ 1 − δ}. Given δ > 0 and n ∈ N, X is said to be P δ, n -smooth if for each f1, f2, . . . , fn ∈ SX∗ there exist 1 ≤ i, j ≤ n, i / j, such that S fi,−fj , δ ∅. X is said to be P n -smooth if it is P δ, n -smooth for some δ > 0, and X is said to be P -smooth if it is P δ, n -smooth for some δ > 0 and some n ∈ N. Proposition 2.21. Let X be a Banach space. Then a X is P(n)-convex if and only if X∗ is P(n)-smooth. b X is P(n)-smooth if and only if X∗ is P(n)-convex. Proof. a Let X be a P ε, n -convex space. Let x∗∗ 1 , . . . , x ∗∗ n ∈ SX∗∗ . We will show that there exist 1 ≤ i, j ≤ n, i / j, such that S x∗∗ i ,−x∗∗ j , ε/4 ∅. Since X is P-convex, it is also reflexive. Therefore x∗∗ 1 j x1 , . . . , x ∗∗ n jj xn for some x1, . . . , xn ∈ SX , where j is the canonical injection from X to X∗∗. By hypothesis, there exist 1 ≤ i, j ≤ n, i / j, such that ‖xi − xj‖ ≤ 2 − ε. Therefore it is enough to prove that { f ∈ BX∗ : f xi ≥ 1 − ε 4 ,−f ( xj ) ≥ 1 − ε 4 } ∅. 2.18 We proceed by contradiction supposing that there exists f ∈ BX∗ such that f xi ≥ 1−ε/4 and −f xj ≥ 1 − ε/4. Then 2 − ε ≥ ∥xi − xj ∥ ≥ fxi − xj ) ≥ 2 − ε 2 , 2.19 which is not possible; consequently X∗ is P ε/4, n -smooth. Now let X be a Banach space such that X∗ is P ε, n -smooth. Let x1, . . . , xn ∈ SX . By hypothesis, there exist 1 ≤ i, j ≤ n, i / j, such that S j xi ,−j xj , ε ∅, that is, for each f ∈ BX∗ we have f xi < 1 − ε or −f xj < 1 − ε. We will see that ‖xi − xj‖ ≤ 2 − ε. We again proceed by contradiction supposing that ‖xi − xj‖ ‖j xi − xj ‖ > 2 − ε. There exists f ∈ SX∗ such that j xi − xj f f xi − f xj > 2 − ε. If f xi < 1 − ε, then 1 ∥f ∥∥xj ∥∥ ≥ −fxj ) > 2 − ε − f xi > 1 2.20 which is not possible. Similarly if −f xj < 1 − ε, we obtain a contradiction. Thus ‖xi − xj‖ ≤ 2 − ε, and consequently X is P ε, n -convex. The proof of b is analogous to the proof of a . Therefore the conditions X is P n -smooth and F n,X > 0 must be equivalent. 3. p-Convex Banach Spaces In this section we introduce the nonuniform version of P-convexity and we call it p-convexity. Abstract and Applied Analysis 7 Definition 3.1. Let X be a Banach space and n ∈ N. X is said to be p n -convex if for any x1, . . . , xn ∈ SX , there exist 1 ≤ i, j ≤ n, i / j, such that ‖xi − xj‖ < 2. X is said to be p-convex if is p n -convex for some n ∈ N.and Applied Analysis 7 Definition 3.1. Let X be a Banach space and n ∈ N. X is said to be p n -convex if for any x1, . . . , xn ∈ SX , there exist 1 ≤ i, j ≤ n, i / j, such that ‖xi − xj‖ < 2. X is said to be p-convex if is p n -convex for some n ∈ N. Kottman defined the concept of P-convexity in terms of the intersection of balls. We will do something similar to give an equivalent definition of p-convexity. It is easy to see that in a normed space any two closed balls of radius 1/2 contained in the unit ball have non empty intersection. If the radius is less than 1/2, for example, in l1 for every n and for every r < 1/2, then there exist n closed balls of radius r so that no two of them intersect. In fact let {ei}i 1 be the canonical basis of l1. Then the closed balls of radius r < 1/2 centered at the points 1/2 ei, i ∈ N are disjoint and contained in the unit ball. However, if X is p n -convex, we will see that for any n points in the unit ball there exists r < 1/2 so that if the n closed balls centered at these n points are contained in the unit ball, there are two different balls with non empty intersection. To prove this we need the following lemma, which was shown in 11 . Lemma 3.2. Let X be a Banach space and x, y ∈ X, x, y / 0. Then ∥∥∥∥ x ‖x‖ − y ∥y ∥ ∥∥∥∥ ≥ 1 min ‖x‖,∥y∥ ∥x − y∥ − ∣‖x‖ − ∥y∥∣. 3.1 Lemma 3.3. X is a p(n)-convex space if and only if for any y1, . . . , yn ∈ BX there exists r ∈ 0, 1/2 such that, if B yi, r ⊂ BX for all i 1, . . . , n, then there are 1 ≤ i, j ≤ n, i / j, so that B ( yi, r ) ∩ Byj, r ) / ∅. 3.2 Proof. Assume that X satisfies condition 3.2 , and let x1, . . . , xn ∈ SX . Let r ∈ 0, 1/2 be the number which satisfies condition 3.2 for x1/2, . . . , xn/2. It is easy to see that B xi/2, r ⊂ BX for each i 1, . . . , n. Therefore there exist 1 ≤ i, j ≤ n, i / j, such that B (xi 2 , r ) ∩ B ( xj 2 , r ) / ∅. 3.3