On 𝑃- and 𝑝-Convexity of Banach Spaces

Abstract

We show that every U-space and every Banach space X satisfying 𝛿_𝑋(1)>0 are P(3)-convex, and we study the nonuniform version of P-convexity, which we call p-convexity.

1. Introduction

Kottman introduced in 1970 the concept of P-convexity in [1]. He proved that every P-convex space is reflexive and also that P-convexity follows from uniform convexity, as well as from uniform smoothness. In this paper we study conditions which guarantee the P-convexity of a Banach space and generalize the result of Kottman concerning uniform convexity in two different ways: every 𝑈-space and every Banach space 𝑋 satisfying 𝛿_𝑋(1)>0 are P(3)-convex. There are many convexity conditions of Banach spaces which have a uniform and also a nonuniform version, for example, strictly convexity is the nonuniform version of uniform convexity, smoothness is the nonuniform version of uniform smoothness, and a 𝑢-space is the nonuniform version of a 𝑈-space, among others. We also define the concept of 𝑝-convexity, which is the nonuniform version of 𝑃-convexity and obtain some interesting results.

2. 𝑃-Convex Banach Spaces

Throughout this paper we adopt the following notation. (𝑋,‖⋅‖) will be a Banach space and when there is no possible confusion, we simply write 𝑋. The unit ball {𝑥∈𝑋∶‖𝑥‖≤1} and the unit sphere {𝑥∈𝑋∶‖𝑥‖=1} are denoted, respectively, by 𝐵_𝑋 and 𝑆_𝑋. 𝐵(𝑦,𝑟) will denote the closed ball with center 𝑦 and radius 𝑟. The topological dual space of 𝑋 is denoted by 𝑋^∗.

2.1. 𝑃-Convexity

The next concept was given by Kottman in [1].

Definition 2.1. Let 𝑋 be a Banach space. For each 𝑛∈ℕ let 𝑃(𝑛,𝑋)=sup{𝑟>0∶thereexist𝑛disjointballsofradius𝑟in𝐵_𝑋}. (2.1) It is easy to see that 𝑃(𝑛,𝑋)≤1/2 for 𝑛≥2.

Definition 2.2. 𝑋 is said to be P-convex if 𝑃(𝑛,𝑋)<1/2 for some 𝑛∈ℕ.

The following lemma was proved in [1].

Lemma 2.3. Let 𝑋 be a Banach space and 𝑛∈ℕ. Then 𝑃(𝑛,𝑋)<1/2 if and only if there exists 𝜀>0 such that for any 𝑥₁,𝑥₂,…,𝑥_𝑛∈𝑆_𝑋min{‖𝑥_𝑖−𝑥_𝑗‖∶1≤𝑖,𝑗≤𝑛,𝑖≠𝑗}≤2−𝜀. (2.2) That is, 𝑋 is P -convex if and only if 𝑋 satisfies condition (2.2) for some 𝑛∈ℕ and some 𝜀>0.

Definition 2.4. Given 𝑛∈ℕ and 𝜀>0 we say that 𝑋 is 𝑃(𝜀,𝑛)-convex if 𝑋 satisfies (2.2). For each 𝑛∈ℕ, 𝑋 is said to be 𝑃(𝑛)-convex if it is 𝑃(𝜀,𝑛)-convex for some 𝜀>0.

2.2. 𝑃-Convexity and the Coefficient of Convexity

In [1], Kottman proved that if 𝑋 is a Banach space satisfying the condition 𝛿_𝑋(2/3)>0, then 𝑋 is P(3)-convex, where 𝛿_𝑋 is the modulus of convexity. In this section we give a result which improves this condition, and we show that this assumption is sharp.

We recall the following concepts introduced by J. A. Clarkson in 1936.

Definition 2.5. The modulus of convexity of a Banach space 𝑋 is the function 𝛿_𝑋∶[0,2]→[0,1] defined by 𝛿_𝑋(𝜀)=inf{1−‖𝑥+𝑦2‖∶𝑥,𝑦∈𝐵_𝑋,‖𝑥−𝑦‖≥𝜀}. (2.3) The coefficient of convexity of a Banach space 𝑋 is the number 𝜀₀(𝑋) defined as 𝜀₀(𝑋)=sup{𝜀∈[0,2]∶𝛿_𝑋(𝜀)=0}. (2.4) We also need the following definition given by R. C. James in 1964.

Definition 2.6. 𝑋 is said to be uniformly nonsquare if there exists 𝛼>0 such that for all 𝜉,𝜂∈𝑆_𝑋min{‖𝜉−𝜂‖,‖𝜉+𝜂‖}≤2−𝛼. (2.5) In order to prove our theorem we need two known results which can be found in [2].

Lemma 2.7 (Goebel-Kirk). Let 𝑋 be a Banach space. For each 𝜀∈[𝜀₀(𝑋),2], one has the equality 𝛿_𝑋(2−2𝛿_𝑋(𝜀))=1−𝜀/2.

Lemma 2.8 (Ullán). Let 𝑋 be a Banach space. For each 0≤𝜀₂≤𝜀₁<2 the following inequality holds: 𝛿_𝑋(𝜀₁)−𝛿_𝑋(𝜀₂)≤(𝜀₁−𝜀₂)/(2−𝜀₁).

Using these lemmas we obtain:

Theorem 2.9. Let X be a Banach space which satisfies 𝛿_X(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∈𝑆_𝑋, and suppose that ‖𝑥−𝑦‖>2−2𝛿_𝑋(𝑡₀) and ‖𝑥−𝑧‖>2−2𝛿_𝑋(𝑡₀). By Lemma 2.7, we have ‖𝑥+𝑦2‖≤1−𝛿(2−2𝛿_𝑋(𝑡₀𝑡))=1−(1−⁰2𝑡)=⁰2. (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 𝛿_𝑋(𝑡₀)=𝛿_𝑋(𝑡₀)−𝛿_𝑋(𝜀₀𝑡(𝑋))≤⁰−𝜀₀(𝑋)2−𝑡₀=2−𝜀₀(𝑋)−1=1−𝑡₀. (2.8) Then ‖𝑦−𝑧‖≤2𝑡₀≤2−2𝛿_𝑋(𝑡₀), and thus 𝑋 is P(3)-convex.
Now consider for each 1<𝑝<∞ the space 𝑙_𝑝,∞ defined as follows. Each element 𝑥={𝑥_𝑖}_𝑖∈𝑙_𝑝 may be represented as 𝑥=𝑥⁺−𝑥⁻, where the respective 𝑖th components of 𝑥⁺ and 𝑥⁻ are given by (𝑥⁺)_𝑖=max{𝑥_𝑖,0} and (𝑥⁻)_𝑖=max{−𝑥_𝑖,0}. Set ‖𝑥‖_𝑝,∞=max{‖𝑥⁺‖_𝑝,‖𝑥⁻‖_𝑝} where ‖⋅‖_𝑝 stands for the 𝑙_𝑝-norm. The space 𝑙_𝑝,∞=(𝑙_𝑝,‖⋅‖_𝑝,∞) satisfies 𝜀₀(𝑙_𝑝,∞)=1 (see [3]). On the other hand let 𝑥₁=𝑒₁−𝑒₃, 𝑥₂=−𝑒₁+𝑒₂, 𝑥₃=−𝑒₂+𝑒₃∈𝑆_𝑙𝑝,∞, where {𝑒_𝑖}_𝑖 is the canonical basis in 𝑙_𝑝. These points satisfy that ‖𝑥_𝑖−𝑥_𝑗‖_𝑝,∞=2, 𝑖≠𝑗. Thus 𝑙_𝑝,∞ 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=𝑙_𝑝,∞ 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 space 𝑋 is the function 𝜌_𝑋∶[0,∞)→[0,∞) defined by 𝜌_𝑋1(𝑡)=sup{2(‖𝑥+𝑡𝑦‖+‖𝑥−𝑡𝑦‖−2)∶𝑥,𝑦∈𝑆_𝑋} (2.9) for each 𝑡≥0. 𝑋 is called uniformly smooth if lim_𝑡→0𝜌_𝑋(𝑡)/𝑡=0.
The proofs of the following lemmas can be found in [4, 5].

Lemma 2.11. For every Banach space 𝑋, one has lim_𝑡→0𝜌_𝑋(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 lim_𝑡→0𝜌_𝑋(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, then 𝑋 is always P(1,5)-convex. Hence the space 𝑋=(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∈𝑆_𝑋,𝑓(𝑥−𝑦)>𝜀,forsome𝑓∈∇(𝑥)⇒‖𝑥+𝑦2‖≤1−𝛿, (2.10) where for each 𝑥∈𝑋∇(𝑥)={𝑓∈𝑆_𝑋^∗∶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∈𝑆_𝑋min{‖𝜉−𝜂‖,‖𝜉+𝜂‖}≤2−𝛼. (2.12) Since 𝑋 is a U-space, for 𝜀=𝛼/2 there exists 𝛿>0 such that 𝑥,𝑦∈𝑆_𝑋𝛼,𝑓(𝑥−𝑦)≥2,forsome𝑓∈∇(𝑥)⇒‖𝑥+𝑦2‖≤1−𝛿. (2.13) We claim that 𝑋 is P (𝛽,3)-convex, where 𝛽=min{𝛼,𝛿}. Indeed, proceeding by contradiction, assume that there exist 𝑥,𝑦,𝑧∈𝑆_𝑋 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+𝑥2‖<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 𝛼2=2𝑓(𝑤)=𝑓(𝑤−𝑥)+𝑓(𝑥−𝑢)+𝑓(𝑢+𝑤)<2𝛼+2−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 𝐵_𝑋 is said to be 𝜀-flat if 𝐴⋂(1−𝜀)𝐵_𝑋=∅. A collection 𝔇 of 𝜀-flats is called complemented if for each pair of 𝜀-flats𝐴 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∶𝐵_𝑋hasacomplementedcollection𝔇of𝜀-flatssuchthatCard(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.

Definition 2.20. Let 𝑋 be a Banach space and 𝛿>0. For each 𝑓,𝑔∈𝑋^∗ set 𝑆(𝑓,𝑔,𝛿)={𝑥∈𝐵_𝑋∶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,…,𝑓_𝑛∈𝑆_𝑋^∗ there exist 1≤𝑖,𝑗≤𝑛,𝑖≠𝑗, such that 𝑆(𝑓_𝑖,−𝑓_𝑗,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 𝑥₁^∗∗,…,𝑥_𝑛^∗∗∈𝑆_𝑋^∗∗. 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 𝑥₁^∗∗=𝚥(𝑥₁),…,𝑥_𝑛^∗∗=𝚥𝚥(𝑥_𝑛) for some 𝑥₁,…,𝑥_𝑛∈𝑆_𝑋, where 𝚥 is the canonical injection from 𝑋 to 𝑋^∗∗. By hypothesis, there exist 1≤𝑖,𝑗≤𝑛, 𝑖≠𝑗, such that ‖𝑥_𝑖−𝑥_𝑗‖≤2−𝜀. Therefore it is enough to prove that {𝑓∈𝐵_𝑋^∗∶𝑓(𝑥_𝑖𝜀)≥1−4,−𝑓(𝑥_𝑗𝜀)≥1−4}=∅. (2.18) We proceed by contradiction supposing that there exists 𝑓∈𝐵_𝑋^∗ such that 𝑓(𝑥_𝑖)≥1−𝜀/4 and −𝑓(𝑥_𝑗)≥1−𝜀/4. Then 2−𝜀≥‖𝑥_𝑖−𝑥_𝑗‖≥𝑓(𝑥_𝑖−𝑥_𝑗𝜀)≥2−2, (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,…,𝑥_𝑛∈𝑆_𝑋. By hypothesis, there exist 1≤𝑖,𝑗≤𝑛, 𝑖≠𝑗, such that 𝑆(𝚥(𝑥_𝑖),−𝚥(𝑥_𝑗),𝜀)=∅, that is, for each 𝑓∈𝐵_𝑋^∗ we have 𝑓(𝑥_𝑖)<1−𝜀 or −𝑓(𝑥_𝑗)<1−𝜀. We will see that ‖𝑥_𝑖−𝑥_𝑗‖≤2−𝜀. We again proceed by contradiction supposing that ‖𝑥_𝑖−𝑥_𝑗‖=‖𝚥(𝑥_𝑖−𝑥_𝑗)‖>2−𝜀. There exists 𝑓∈𝑆_𝑋^∗ such that 𝚥(𝑥_𝑖−𝑥_𝑗)(𝑓)=𝑓(𝑥_𝑖)−𝑓(𝑥_𝑗)>2−𝜀. If 𝑓(𝑥_𝑖)<1−𝜀, then1=‖𝑓‖‖𝑥_𝑗‖≥−𝑓(𝑥_𝑗)>2−𝜀−𝑓(𝑥_𝑖)>1 (2.20) which is not possible. Similarly if −𝑓(𝑥_𝑗)<1−𝜀, we obtain a contradiction. Thus ‖𝑥_𝑖−𝑥_𝑗‖≤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.

Definition 3.1. Let 𝑋 be a Banach space and 𝑛∈ℕ. 𝑋 is said to be p (𝑛)-convex if for any 𝑥₁,…,𝑥_𝑛∈𝑆_𝑋, there exist 1≤𝑖,𝑗≤𝑛, 𝑖≠𝑗, such that ‖𝑥_𝑖−𝑥_𝑗‖<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 {𝑒_𝑖}^∞_𝑖=1 be the canonical basis of 𝑙₁. Then the closed balls of radius 𝑟<1/2 centered at the points (1/2)𝑒_𝑖, 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−𝑦‖𝑥‖1‖𝑦‖‖≥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,…,𝑦_𝑛∈𝐵_𝑋 there exists 𝑟∈(0,1/2) such that, if 𝐵(𝑦_𝑖,𝑟)⊂𝐵_𝑋 for all 𝑖=1,…,𝑛, then there are 1≤𝑖,𝑗≤𝑛, 𝑖≠𝑗, so that 𝐵(𝑦_𝑖,𝑟)∩𝐵(𝑦_𝑗,𝑟)≠∅. (3.2)

Proof. Assume that 𝑋 satisfies condition (3.2), and let 𝑥₁,…,𝑥_𝑛∈𝑆_𝑋. Let 𝑟∈(0,1/2) be the number which satisfies condition (3.2) for 𝑥₁/2,…,𝑥_𝑛/2. It is easy to see that 𝐵(𝑥_𝑖/2,𝑟)⊂𝐵_𝑋 for each 𝑖=1,…,𝑛. Therefore there exist 1≤𝑖,𝑗≤𝑛, 𝑖≠𝑗, such that 𝑥𝐵(^𝑖2𝑥,𝑟)∩𝐵(^𝑗2,𝑟)≠∅. (3.3) Let 𝑥𝑦∈𝐵(^𝑖2𝑥,𝑟)∩𝐵(^𝑗2,𝑟). (3.4) We have ‖𝑥_𝑖−𝑥_𝑗2𝑥‖≤‖^𝑖2𝑥−𝑦‖+‖^𝑗2−𝑦‖<2𝑟<1, (3.5) and thus 𝑋 is p(𝑛)-convex. Now we suppose that there exist 𝑦₁,…,𝑦_𝑛∈𝐵_𝑋 such that for any 𝜌∈(0,1/2) we have 𝐵(𝑦_𝑖,12−𝜌)⊂𝐵_𝑋 (3.6) for all 𝑖=1,…,𝑛, and 𝐵(𝑦_𝑖,12−𝜌)∩𝐵(𝑦_𝑗,12−𝜌)=∅, (3.7) for all 𝑖,𝑗=1,…,𝑛, 𝑖≠𝑗. We verify that 𝑋 is not p(𝑛)-convex in four steps.(a)Take‖𝑦_𝑖−𝑦_𝑗‖>1−2𝜌 for any 𝑖,𝑗=1,…,𝑛, 𝑖≠𝑗.(b)Take1/2−3𝜌<‖𝑦_𝑖‖≤1/2+𝜌, for all 𝑖=1,…,𝑛. To verify this claim we note that ‖𝑦_𝑖/‖𝑦_𝑖‖−𝑦_𝑖‖≥1/2−𝜌 for all 𝑖, because if ‖𝑦_𝑖/‖𝑦_𝑖‖−𝑦_𝑖‖<1/2−𝜌 for some 𝑖, then 𝑦_𝑖/‖𝑦_𝑖‖∈int𝐵(𝑦_𝑖,1/2−𝜌)⊂int𝐵_𝑋, which is not possible. Hence, as ‖𝑦_𝑖/‖𝑦_𝑖‖−𝑦_𝑖‖=1−‖𝑦_𝑖‖, it follows that ‖𝑦_𝑖‖=1−‖𝑦_𝑖/‖𝑦_𝑖‖−𝑦_𝑖‖≤1/2+𝜌, for each 𝑖=1,…,𝑛. Now, if ‖𝑦_𝑖‖≤1/2−3𝜌 for some 𝑖, we have by (a) that for any 𝑗≠𝑖, 1−2𝜌<‖𝑦_𝑖−𝑦_𝑗‖≤‖𝑦_𝑖‖+‖𝑦_𝑗‖≤(1/2−3𝜌)+(1/2+𝜌)=1−2𝜌 which is not possible.(c)Take|‖𝑦_𝑖‖−‖𝑦_𝑗‖|<4𝜌, for any 𝑖,𝑗=1,…,𝑛, 𝑖≠𝑗. Indeed, by (b) we get −4𝜌=(1/2−3𝜌)−(1/2+𝜌)<‖𝑦_𝑖‖−‖𝑦_𝑗‖<(1/2+𝜌)−(1/2−3𝜌)=4𝜌.(d)From (a), (b), (c), and by Lemma 3.2, we have ‖𝑦_𝑖‖𝑦_𝑖‖−𝑦_𝑗‖𝑦_𝑗‖1‖≥‖𝑦_𝑖‖(‖𝑦_𝑖−𝑦_𝑗‖−|‖𝑦_𝑖‖−‖𝑦_𝑗‖|)>2−16𝜌1+2𝜌 (3.8) for any 𝑖,𝑗=1,…,𝑛, 𝑖≠𝑗. Since 𝜌>0 is arbitrary, as 𝜌→0, we obtain ‖𝑦_𝑖/‖𝑦_𝑖‖−𝑦_𝑗/‖𝑦_𝑗‖‖=2, for all 𝑖,𝑗=1,…,𝑛, 𝑖≠𝑗, and thus 𝑋 is not p(n)-convex.

Next we give some examples of spaces which are not p-convex. The first is not reflexive and the last one is superreflexive.

Example 3.4. 𝑐₀, and consequently, 𝐶[0,1] and 𝑙_∞ are not p-convex spaces. Indeed, let {𝑒_𝑖}^∞_𝑖=1 be the canonical basis in 𝑐₀. For each 𝑛∈ℕ we define 𝑢_𝑖=∑^𝑛_𝑗=1𝜆_𝑖,𝑗𝑒_𝑗, where 𝜆_𝑖,𝑗=1 if 𝑗≠𝑖, 𝜆_𝑖,𝑖=−1, and 𝑖=1,…,𝑛. Clearly 𝑢₁,…,𝑢_𝑛∈𝑆_𝑐0, and for each 𝑖≠𝑗 we have ‖𝑢_𝑖−𝑢_𝑗‖_∞=2.

Example 3.5. Let 𝑋 denote the space obtained by renorming 𝑙₂ as follows. For 𝑥=(𝑥_𝑖)_𝑖∈ℕ∈𝑙₂ set ‖|𝑥|‖=max{sup_𝑖,𝑗|𝑥_𝑖−𝑥_𝑗|,(^∞_𝑖=1𝑥_𝑖²)^1/2}. (3.9) Then ‖𝑥‖≤‖|𝑥|‖≤√2‖𝑥‖, where ‖⋅‖ stands for the 𝑙₂-norm and 𝑋 is superreflexive. On the other hand, the canonical basis {𝑒_𝑛}_𝑛 in 𝑙₂ satisfies ‖𝑒_𝑖−𝑒_𝑗‖_∞=2 for each 𝑖≠𝑗. Thus 𝑋 is not p-convex.

Now we will mention several properties that imply p-convexity.

Recall the following concepts. Let 𝑋 be a Banach space. 𝑋 is said to be a u-space if it satisfies the following implication: 𝑥,𝑦∈𝑆_𝑋,‖𝑥+𝑦2‖=1⇒∇(𝑥)=∇(𝑦). (3.10)X is said to be smooth if for any 𝑥∈𝑆_𝑋, there exists a unique 𝑓∈𝑆_𝑋^∗ such that 𝑓(𝑥)=1. That is, for each 𝑥∈𝑆_𝑋, ∇(𝑥) contains a single point. X is called strictly convex if the following implication holds: ∀𝑥,𝑦∈𝐵_𝑋∶𝑥≠𝑦⇒‖𝑥+𝑦2‖<1. (3.11)

Proposition 3.6. Every smooth space, every strictly convex space and every u-space are p(3)-convex space.

Proof. Every smooth space and every strictly convex space are u-space. It suffices to show that p(3)-convexity follows from being u-space. If 𝑋 is a u-space, then for any 𝑥,𝑦∈𝑆_𝑋 the following inequality holds: min{‖𝑥−𝑦‖,‖𝑥+𝑦‖}<2. Indeed, if we suppose that there exist 𝑥,𝑦∈𝑆_𝑋 such that ‖𝑥+𝑦‖=‖𝑥−𝑦‖=2, then ∇(𝑥)=∇(𝑦) and ∇(𝑥)=∇(−𝑦), which is not possible. Suppose that 𝑋 is not p(3)-convex, and there exist 𝑥,𝑦,𝑧∈𝑆_𝑋 so that ‖𝑥−𝑦‖=‖𝑦−𝑧‖=‖𝑧−𝑥‖=2. Since (1/2)‖𝑥−𝑦‖=(1/2)‖𝑦−𝑧‖=1,we have ∇(𝑥)=∇(−𝑦)=∇(𝑧). Let 𝑓∈∇(−𝑦); then 𝑓(𝑥+𝑧)≤‖𝑥+𝑧‖<2, and 2=𝑓(𝑥)+𝑓(−𝑦)=𝑓(𝑥+𝑧)−𝑓(𝑧)+𝑓(−𝑦)=𝑓(𝑥+𝑧)<2. (3.12) Thus 𝑋 is p(3)-convex.

Obviously P-convexity implies p-convexity; however, a p-convex space is not necessarily P-convex, even if the space is reflexive as the following example shows.

Example 3.7. Let {𝑟_𝑘}^∞_𝑘=1 be a sequence of real numbers such that 𝑟_𝑘>1 for each 𝑘∈ℕ and 𝑟_𝑘↓1, when 𝑘→∞. Consider the space 𝑋=∑^∞_𝑘=1⊕₂𝑙_𝑟𝑘. It is known that this space is strictly convex, hence it is also p( 3)-convex. It is also known that 𝑋 is reflexive. However 𝑋 is not P-convex. Indeed, let 𝜀>0. We choose 𝑘∈ℕ such that 2−𝜀<2^1/𝑟𝑘. If {𝑒_𝑖}^∞_𝑖=1 is the canonical basis of 𝑙_𝑟𝑘, we have that ‖𝑒_𝑖−𝑒_𝑗‖_𝑟𝑘=2^1/𝑟𝑘>2−𝜀 for all 𝑖,𝑗∈ℕ, 𝑖≠𝑗, and hence 𝑋 is not a P-convex space.

We have obtained a result which shows a strong relation between P-convexity and p-convexity with respect to the ultrapower of Banach spaces. We recall the definition and some results regarding ultrapowers which can be found in [4].

A filter 𝔘 on 𝐼 is called an ultrafilter on 𝐼 if 𝔘 is a maximal element from 𝒫 with respect to the set inclusion. 𝔘 is an ultrafilter on 𝐼 if and only if for all 𝐴⊂𝐼 either 𝐴∈𝔘 or 𝐼⧵𝐴∈𝔘. Let {𝑋_𝑖}_𝑖∈𝐼 be a family of Banach spaces, and let𝑙_∞(𝑋_𝑖)={{𝑥_𝑖}_𝑖∈𝐼∈_𝑖∈𝐼𝑋_𝑖∶sup{‖𝑥_𝑖‖_𝑋𝑖∶𝑖∈𝐼}<∞}. (3.13) If we define ‖{𝑥_𝑖}_𝑖∈𝐼‖_∞=sup{‖𝑥_𝑖‖_𝑋𝑖∶𝑖∈𝐼} for each {𝑥_𝑖}_𝑖∈𝐼∈𝑙_∞(𝑋_𝑖), then ‖⋅‖_∞ defines a norm in _∞(𝑋_𝑖), and (𝑙_∞(𝑋_𝑖),‖⋅‖_∞) is a Banach space. If 𝔘 is a free ultrafilter on 𝐼, then for each {𝑥_𝑖}_𝑖∈𝐼∈𝑙_∞(𝑋_𝑖) we have lim_𝔘𝑥_𝑖 always exists and is unique. Let 𝔘 be an ultrafilter on 𝐼, and define 𝒩_𝔘={{𝑥_𝑖}∈𝑙_∞(𝑋_𝑖)∶lim_𝔘‖𝑥_𝑖‖=0}. (3.14)𝒩_𝔘 is a closed subspace of 𝑙_∞(𝑋_𝑖). The ultraproduct of {𝑋_𝑖}_𝑖∈𝐼 with respect to the ultrafilter 𝔘 on 𝐼 is the quotient space 𝑙_∞(𝑋_𝑖)/𝒩_𝔘 equipped with the quotient norm, which is denoted by {𝑋_𝑖}_𝔘 and its elements by {𝑥_𝑖}_𝔘. If 𝑋_𝑖=𝑋 for all 𝑖∈𝐼, then {𝑋}_𝔘={𝑋_𝑖}_𝔘 is called the ultrapower of 𝑋. The quotient norm in {𝑋_𝑖}_𝔘, ‖{𝑥_𝑖}_𝔘‖=inf{‖{𝑥_𝑖+𝑦_𝑖}_𝑖‖_∞∶{𝑦_𝑖}_𝑖∈𝒩_𝔘}, (3.15) satisfies the equality ‖{𝑥_𝑖}_𝔘‖=lim_𝔘‖𝑥_𝑖‖_X𝑖,foreach{𝑥_𝑖}_𝔘∈{𝑋_𝑖}_𝔘. (3.16) If 𝔘 is nontrivial, then 𝑋 can be embedded into {𝑋}_𝔘 isometrically. We will write 𝑋_𝑖 instead of {𝑋_𝑖}_𝔘 and 𝑥 instead of {𝑥_𝑖}_𝔘 unless we need to specify the ultrafilter we are talking about.

It is known that 𝑋 is uniformly convex if and only if 𝑋 is strictly convex, 𝑋 is uniformly smooth if and only if 𝑋 is smooth, and 𝑋 is a U-space if and only if 𝑋 is a u-space (see [12]). Similarly we obtain the following result.

Theorem 3.8. Let 𝑋 be a Banach space and 𝑚∈ℕ. The following are equivalent:

(a)𝑋 is P(𝑚)-convex.(b)𝑋 is P(𝑚)-convex,(c)𝑋 is p(𝑚)-convex,

Proof. (𝑎)⇒(𝑏). Let {𝑥_𝑖^(𝑛)}_𝑛∈𝑥_𝑖, 𝑥_𝑖∈𝑆_𝑋, 𝑖=1,…,𝑚. Since lim_𝔘‖𝑥_𝑖^(𝑛)‖_𝑋=‖𝑥_𝑖‖_𝑋=1 for all 𝑖, there exists a subsequence {𝑥^(𝑛𝑘)_𝑖}_𝑘 of {𝑥_𝑖^(𝑛)}_𝑛 such that lim_𝑘→∞‖𝑥^(𝑛𝑘)_𝑖‖_𝑋=1 and ‖𝑥^(𝑛𝑘)_𝑖‖_𝑋>0, for all 𝑘∈ℕ. Define 𝑦^(𝑛𝑘)_𝑖=𝑥^(𝑛𝑘)_𝑖‖𝑥^(𝑛𝑘)_𝑖‖_𝑋,Γ_𝑖,𝑗={𝑘∈ℕ∶‖𝑦^(𝑛𝑘)_𝑖−𝑦^(𝑛𝑘)_𝑗‖_𝑋≤2−𝜀}, (3.17) for each 𝑖,𝑗=1,…,𝑚, 𝑖≠𝑗. We verify that there exist 1≤𝑖,𝑗≤𝑚, 𝑖≠𝑗, such that Γ_𝑖,𝑗∈𝔘. We proceed by contradiction assuming that, Γ_𝑖,𝑗∉𝔘 for all 𝑖≠𝑗. Hence ℕ⧵Γ_𝑖,𝑗∈𝔘 for all 𝑖≠𝑗, and consequently ℕ⧵(⋃_𝑖≠𝑗Γ_𝑖,𝑗)≠∅, therefore there exists 𝑘₀∈ℕ⧵(⋃_𝑖≠𝑗Γ_𝑖,𝑗). Thus we have ‖𝑦^(𝑛𝑘0)_𝑖−𝑦^(𝑛𝑘0)_𝑗‖>2−𝜀 for each 𝑖≠𝑗, and 𝑋 is not P(𝑚)-convex, which is a contradiction. Therefore there exist 1≤𝑖,𝑗≤𝑚, 𝑖≠𝑗, such that Γ_𝑖,𝑗∈𝔘, and hence lim_𝔘‖𝑦^(𝑛𝑘)_𝑖−𝑦^(𝑛𝑘)_𝑗‖_𝑋≤2−𝜀. Finally, note that ‖𝑥^(𝑛𝑘)_𝑖−𝑥^(𝑛𝑘)_𝑗‖_𝑋≤‖𝑥^(𝑛𝑘)_𝑖−𝑦^(𝑛𝑘)_𝑖‖_𝑋+‖𝑥^(𝑛𝑘)_𝑗−𝑦^(𝑛𝑘)_𝑗‖_𝑋+‖𝑦^(𝑛𝑘)_𝑖−𝑦^(𝑛𝑘)_𝑗‖_𝑋=|1−‖𝑥^(𝑛𝑘)_𝑖‖_𝑋|+|1−‖𝑥^(𝑛𝑘)_𝑖‖_𝑋|+‖𝑦^(𝑛𝑘)_𝑖−𝑦^(𝑛𝑘)_𝑗‖_𝑋,‖𝑥_𝑖−𝑥_𝑗‖_𝑋=lim_𝔘‖𝑥_𝑖^(𝑛)−𝑥_𝑗^(𝑛)‖_𝑋=lim_𝔘‖𝑥^(𝑛𝑘)_𝑖−𝑥^(𝑛𝑘)_𝑗‖_𝑋≤lim_𝔘|1−‖𝑥^(𝑛𝑘)_𝑖‖_𝑋|+lim_𝔘|1−‖𝑥^(𝑛𝑘)_𝑖‖_𝑋|+lim_𝔘‖𝑦^(𝑛𝑘)_𝑖−𝑦^(𝑛𝑘)_𝑗‖_𝑋≤2−𝜀. (3.18) Therefore 𝑋 is P(𝑚)-convex.
(𝑏)⇒(𝑐) is obvious.
(𝑐)⇒(𝑎). Suppose that 𝑋 is not P(𝑚)-convex. Hence for any 𝑛∈ℕ there exist 𝑥₁^(𝑛),…,𝑥_𝑚^(𝑛)∈𝑆_𝑋 such that ‖𝑥_𝑖^(𝑛)−𝑥_𝑗^(𝑛)‖_𝑋>2−1/𝑛 for all 𝑖,𝑗=1,…,𝑚, 𝑖≠𝑗. Define 𝑥_𝑖={𝑥_𝑖^(𝑛)}_𝔘 for each 𝑖=1,…,𝑚. Clearly 𝑥_𝑖∈𝑆_𝑋 for all 𝑖, because ‖𝑥_𝑖‖_𝑋=lim_𝔘‖𝑥_𝑖^(𝑛)‖_𝑋=1, and also,‖𝑥_𝑖−𝑥_𝑗‖_𝑋=lim_𝔘‖𝑥_𝑖^(𝑛)−𝑥_𝑗^(𝑛)‖_𝑋=lim_𝑛→∞‖𝑥_𝑖^(𝑛)−𝑥_𝑗^(𝑛)‖_𝑋=2, (3.19) for each 𝑖≠𝑗. Hence 𝑋 is not p(𝑚)-convex.

By the above theorem we can deduce the following known result.

Corollary 3.9. If 𝑋 is P -convex, then 𝑋 is superreflexive.

Proof. If 𝑋 is P-convex, then 𝑋 is P-convex and therefore is reflexive. However in ultrapower reflexivity and superreflexivity are equivalent, hence 𝑋 is superreflexive, and consequently 𝑋 is superreflexive.

Now we turn our attention to some results regarding the p-convexity and the P-convexity of quotient spaces. To prove them we need the following concept.

Definition 3.10. A subspace 𝑌 of a normed space 𝑋 is said to be proximinal if for all 𝑥∈𝑋 there exists 𝑦∈𝑌 such that 𝑑(𝑥,𝑌)=‖𝑥−𝑦‖.

It is easy to see that every proximinal subspace 𝑌 of a Banach space 𝑋 is closed.

Proposition 3.11. If 𝑋 is p(𝑛)-convex and 𝑌 is a proximinal subspace of 𝑋, then 𝑋/𝑌 is p (𝑛)-convex.

Proof. Let 𝑞∶𝑋→𝑋/𝑌 be the quotient function. By the proximinality of 𝑌 we have 𝑞(𝐵_𝑋)=𝐵_𝑋/𝑌. Let 𝑥₁,…,𝑥_𝑛∈𝑆_𝑋/𝑌 and 𝑥₁,…,𝑥_𝑛∈𝑆_𝑋 such that 𝑥_𝑖=𝑞(𝑥_𝑖). Since 𝑋 is p(𝑛)-convex, there exist 1≤𝑖,𝑗≤𝑛, 𝑖≠𝑗, such that ‖𝑥_𝑖−𝑥_𝑗‖<2, and consequently ‖𝑥_𝑖−𝑥_𝑗‖<2.

Corollary 3.12. Let 𝑋 be p(𝑛)-convex and reflexive. If 𝑌 is a closed subspace of 𝑋, then 𝑋/𝑌 is p(𝑛)-convex.

Proof. It is shown in [13] that a Banach space X is reflexive if and only if each closed subspace of 𝑋 is proximinal, and thus the corollary is a consequence of Proposition 3.11.

Similarly we can prove that if 𝑋 is P(𝜀,𝑛)-convex and 𝑌 is a closed subspace of 𝑋, then 𝑋/𝑌 is P(𝜀,𝑛)-convex.

We obtain two results involving 𝜓-direct sums of p-convex spaces. Next we will define these sums as in [14] by Saito, et al.

Definition 3.13. Set Ψ={𝜓∶[0,1]→ℝ∣𝜓isacontinuousconvexfunction,max{1−𝑡,𝑡}≤𝜓(𝑡)≤1,forall0≤𝑡≤1.}
Let (𝑋,‖⋅‖_𝑋) and (𝑌,‖⋅‖_𝑌) be Banach spaces. For each 𝜓∈Ψ, one defines the norm ‖⋅‖_𝜓 in 𝑋⊕𝑌 as ‖(0,0)‖_𝜓=0 and for each (𝑥,𝑦)≠(0,0)‖(𝑥,𝑦)‖_𝜓=(‖𝑥‖_𝑋+‖𝑦‖_𝑌)𝜓(‖𝑦‖_𝑌‖𝑥‖_𝑋+‖𝑦‖_𝑌). (3.20)

In [15] it is shown that (𝑋⊕𝑌,‖⋅‖_𝜓) is a Banach space, denoted by 𝑋⊕_𝜓𝑌 called the 𝜓-direct and sum of 𝑋 and 𝑌.

The proof of the following theorem is similar to the proof of Theorem 3.5 in [16], which shows the corresponding result for P-convex spaces.

Theorem 3.14. Let 𝑋 and 𝑌 be Banach spaces and 𝜓∈Ψ. Then 𝑋⊕_𝜓𝑌 is p-convex if and only if 𝑋 and 𝑌 are p-convex.

In [17] there is a theorem stating several equivalent conditions for strict convexity. We prove a similar result for p-convexity.

Lemma 3.15. Let 𝑋 be a Banach space. The next assertions are equivalent.(a)𝑋 is p(n)-convex.(b)For any 𝑞∈(1,∞) and for any 𝑥₁,…,𝑥_𝑛∈𝑋, not all zero, there exist 1≤𝑖,𝑗≤𝑛, 𝑖≠𝑗, such that ‖𝑥_𝑖−𝑥_𝑗‖<2^{(𝑞−1)/𝑞}(‖𝑥_𝑖‖^𝑞+‖𝑥_𝑗‖^𝑞)^1/𝑞.(c)For some 𝑞∈(1,∞) and for any 𝑥₁,…,𝑥_𝑛∈𝑋, not all zero, there exist 1≤𝑖,𝑗≤𝑛, 𝑖≠𝑗, such that ‖𝑥_𝑖−𝑥_𝑗‖<2^{(𝑞−1)/𝑞}(‖𝑥_𝑖‖^𝑞+‖𝑥_𝑗‖^𝑞)^1/𝑞.

Proof. The implications (𝑏)⇒(𝑐)⇒(𝑎) are immediate. We verify (𝑎)⇒(𝑏). Let 𝑞∈(1,∞) and 𝑥₁,…,𝑥_𝑛∈𝑋, not all zero. If 𝑥_𝑗=0 and 𝑥_𝑖≠0 for some 1≤𝑖,𝑗≤𝑛, then it is clear that ‖𝑥_𝑖−𝑥_𝑗‖<2^{(𝑞−1)/𝑞}(‖𝑥_𝑖‖^𝑞+‖𝑥_𝑗‖^𝑞)^1/𝑞. Suppose that 𝑥₁,…,𝑥_𝑛∈𝑋⧵{0}. There exist 1≤𝑖,𝑗≤𝑛, 𝑖≠𝑗, such that ‖𝑥_𝑖‖𝑥_𝑖‖−𝑥_𝑗‖𝑥_𝑗‖‖<2. (3.21) If ‖𝑥_𝑗‖≤‖𝑥_𝑖‖ by Lemma 3.2 we get ‖𝑥_𝑖−𝑥_𝑗‖≤‖𝑥_𝑗𝑥‖‖^𝑖‖𝑥_𝑖‖−𝑥_𝑗‖𝑥_𝑗‖‖+‖𝑥_𝑖‖+‖𝑥_𝑗‖<‖𝑥_𝑖‖+‖𝑥_𝑗‖. (3.22) As the function 𝑡↦𝑡^𝑞 is convex we obtain that ‖𝑥_𝑖−𝑥_𝑗2‖^𝑞<(‖𝑥_𝑖‖+‖𝑥_𝑗‖2)^𝑞≤12(‖𝑥_𝑖‖^𝑞+‖𝑥_𝑗‖^𝑞). (3.23) Thus ‖𝑥_𝑖−𝑥_𝑗‖<2^{(𝑞−1)/𝑞}(‖𝑥_𝑖‖^𝑞+‖𝑥_𝑗‖^𝑞)^1/𝑞.

Proposition 3.16. Let {𝑋_𝑖}_𝑖∈𝐼 be a family of p(n)-convex spaces, where the index set 𝐼≠∅ has any cardinality. Then the space 𝑋=𝑙_𝑞(𝑋_𝑖) (1<𝑞<∞) is p(n)-convex.

Proof. Let 𝑥^(𝑘)={𝑥_𝑖^(𝑘)}_𝑖∈𝐼∈𝑋, 1≤𝑘≤𝑛, not all zero. Let 𝑖₀∈𝐼 be such that 𝑥_𝑖^(𝑘)₀≠0, for some 𝑘∈{1,…,𝑛}. As 𝑋_𝑖0 is a p(n)-convex space, we have by the preceding lemma that there exist 1≤𝑙,𝑚≤𝑛 such that ‖𝑥_𝑖^(𝑙)₀−𝑥_𝑖^(𝑚)₀‖^𝑞<2^𝑞−1(‖𝑥_𝑖^(𝑙)₀‖^𝑞+‖𝑥_𝑖^(𝑚)₀‖^𝑞). (3.24) By the above we obtain ‖𝑥^(𝑙)−𝑥^(𝑚)‖^𝑞_𝑞=_𝑖∈𝐼‖𝑥_𝑖^(𝑙)−𝑥_𝑖^(𝑚)‖^𝑞<_𝑖∈𝐼2^𝑞−1(‖𝑥_𝑖^(𝑙)‖^𝑞+‖𝑥_𝑖^(𝑚)‖^𝑞)=2^𝑞−1(‖𝑥^(𝑙)‖^𝑞_𝑞+‖𝑥^(𝑚)‖^𝑞_𝑞). (3.25) Therefore, by the previous lemma, 𝑋 is p(n)-convex.

Acknowledgments

The author is grateful to Helga Fetter and to Berta Gamboa for their valuable remarks and suggestions which led to substantial improvements in the paper. The author is partially supported by CONACyT, CIMAT, and SEP-CONACyT Grant 102380.