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A geometrical constant and normal normal structure in Banach Spaces
Journal of Inequalities and Applications volume 2011, Article number: 16 (2011)
Abstract
Recently, we introduced a new coefficient as a generalization of the modulus of smoothness and Pythagorean modulus such as J_{ X } , _{ p }(t). In this paper, We can compute the constant J_{ X }, _{ p }(1) under the absolute normalized norms on ℝ^{2} by means of their corresponding continuous convex functions on [0, 1]. Moreover, some sufficient conditions which imply uniform normal structure are presented.
2000 Mathematics Subject Classification: 46B20.
1. Introduction and preliminaries
We assume that X and X* stand for a Banach space and its dual space, respectively. By S_{ X } and B_{ X } we denote the unit sphere and the unit ball of a Banach space X, respectively. Let C be a nonempty bounded closed convex subset of a Banach space X. A mapping T : C → C is said to be nonexpansive provided the inequality
holds for every x, y ∈ C. A Banach space X is said to have the fixed point property if every nonexpansive mapping T : C → C has a fixed point, where C is a nonempty bounded closed convex subset of a Banach space X.
Recall that a Banach space X is called uniformly nonsquare if there exists δ > 0 such that x + y/2 ≤ 1  δ or x  y/2 ≤ 1  δ whenever x, y ∈ S_{ X } . A bounded convex subset K of a Banach space X is said to have normal structure if for every convex subset H of K that contains more than one point, there exists a point x_{0} ∈ H such that
A Banach space X is said to have uniform normal structure if there exists 0 < c < 1 such that for any closed bounded convex subset K of X that contains more than one point, there exists x_{0} ∈ K such that
It was proved by Kirk that every reflexive Banach space with normal structure has the fixed point property.
There are several constants defined on Banach spaces such as the James [1] and von NeumannJordan constants [2]. It has been shown that these constants are very useful in geometric theory of Banach spaces, which enable us to classify several important concept of Banach spaces such as uniformly nonsquareness and uniform normal structure [3–8]. On the other hand, calculation of the constant for some concrete spaces is also of some interest [2, 5, 6, 9].
Recently, we introduced a new coefficient as a generalization of the modulus of smoothness and Pythagorean modulus such as J_{ X }, _{ p }(t).
Definition 1.1. Let x ∈ S_{ X } , y ∈ S_{ X } . For any t > 0, 1 ≤ p < ∞ we set
Some basic properties of this new coefficient are investigated in [6]. In particular, we compute the new coefficient in the Banach spaces l_{ r } , L_{ r } , l_{1}, ∞ and give rough estimates of the constant in some concrete Banach spaces. In fact, the constant J_{X, p}(1) is also important from the below Corollary in [6].
Corollary 1.2. If . Then R(X) < 2, where R(X) and ω(X) stand for GarcíaFalset constant and the coefficient of weak orthogonality, respectively (see [10, 11]). It is well known that a reflexive Banach space X with R(X) < 2 enjoys the fixed property (see [10]).
In this paper, we compute the constant J_{ X }, _{ p }(1) under the absolute normalized norms on ℝ^{2}, and give exact values of the constant J_{ X }, _{ p }(1) in some concrete Banach spaces. Moreover, some sufficient conditions which imply uniform normal structure are presented.
Recall that a norm on ℝ^{2} is called absolute if (z, w) = (z, w) for all z, w ∈ ℝ and normalized if (1,0) = (0,1). Let N_{ α } denote the family of all absolute normalized norms on ℝ^{2}, and let Ψ denote the family of all continuous convex functions on [0, 1] such that ψ (1) = ψ (0) = 1 and max{1  s, s} ≤ ψ(s) ≤ 1(0 ≤ s ≤ 1). It has been shown that N_{ α } and Ψ are a onetoone correspondence in view of the following proposition in [12].
Proposition 1.3. If ·∈ N_{ α } , then ψ(s) = (1  s, s) ∈ Ψ. On the other hand, if ψ(s) ∈Ψ, defined a norm · _{ ψ } as
then the norm ·_{ ψ }∈ N_{ α } .
A simple example of absolute normalized norm is usual l_{ r } (1 ≤ r ≤ ∞) norm. From Proposition 1.3, one can easily get the corresponding function of the l_{ r } norm:
Also, the above correspondence enable us to get many nonl_{ r } norms on ℝ^{2}. One of the properties of these norms is stated in the following result.
Proposition 1.4. Let ψ, φ ∈ Ψ and φ ≤ ψ. Put , then
The Cesà ro sequence space was defined by Shue [13] in 1970. It is very useful in the theory of matrix operators and others. Let l be the space of real sequences.
For 1 < p < ∞, the Cesà ro sequence space ces_{ p } is defined by
The geometry of Cesà ro sequence spaces have been extensively studied in [14–16]. Let us restrict ourselves to the twodimensional Cesà ro sequence space which is just ℝ^{2} equipped with the norm defined by
2. Geometrical constant J_{ X, p }(1) and absolute normalized norm
In this section, we give a simple method to determine and estimate the constant J_{ X, p } (1) of absolute normalized norms on ℝ^{2}. For a norm  ·  on ℝ^{2}, we write J_{ X, p } (1)( · ) for J_{ X, p } (1)(ℝ^{2},  · ). The following is a direct result of Proposition 2.4 in [6].
Proposition 2.1. Let X be a nontrivial Banach space. Then
Proposition 2.2. Let X be the space l_{ r } or L_{ r } [0, 1] with dimX ≥ 2 (see [6])

(1)
Let 1 < r ≤ 2 and 1/r + 1/r' = 1. Then for all t > 0
if 1 < p < r' then .
if r' ≤ p < ∞ then , for some K ≥ 1.

(2)
Let 2 ≤ r < ∞, 1 ≤ p < ∞ and h = max{r, p}. Then
Proposition 2.3. Let φ ∈ Ψ and ψ(s) = φ (1  s). Then
Proof. For any x = (a, b) ∈ ℝ^{2} and a ≠ 0, b ≠ 0, put . Then
Consequently, we have
We now consider the constant J_{ X, p } (1) of a class of absolute normalized norms on ℝ^{2}. Now let us put
Theorem 2.4. Let ψ ∈ Ψ and ψ ≤ ψ_{ r } (2 ≤ r < ∞). If the function attains its maximum at s = 1/ 2 and r ≥ p, then
Proof. By Proposition 1.4, we have  ·  _{ ψ } ≤  ·  _{ r } ≤ M_{1} · _{ ψ }. Let x, y ∈ X, (x, y) ≠ (0, 0), where X = ℝ^{2}. Then
from the definition of J_{ X, p } (t), implies that
Note that r ≥ p and the function attains its maximum at s = 1/ 2, i.e.,. From Proposition 2.2, implies that
On the other hand, let us put x = (a, a), y = (a, a), where . Hence x _{ ψ } = y _{ ψ } = 1, and
From (1) and (2), we have
Theorem 2.5. Let ψ ∈ Ψ and ψ ≥ ψ_{ r } (1 ≤ r ≤ 2). If the function attains its maximum at s = 1/ 2 and 1 ≤ p < r', then
Proof. By Proposition 1.4, we have  ·  _{ r } ≤  ·  _{ ψ } ≤ M_{2} ·  _{ r }. Let x, y ∈ X, (x, y) ≠ (0, 0), where X = ℝ^{2}. Then
From the definition of J_{ X, p } (t), it implies that
note that 1 ≤ p < r' and the function attains its maximum at s = 1/ 2, i.e.,. From Proposition 2.2, it implies that
On the other hand, let us put x = (1, 0), y = (0, 1). Then x _{ ψ } = y _{ ψ } = 1, and
From (3) and (4), we have
Lemma 2.6 (see [6]). Let  ·  and . be two equivalent norms on a Banach space. If a. ≤  ·  ≤ b. (b ≥ a > 0), then
Example 2.7. Let X = ℝ^{2} with the norm
Then
Proof. It is very easy to check that x = max{x_{2}, λx_{1}} ∈ ℕ _{ α } and its corresponding function is
Therefore,
Since ψ_{2}(s) attains minimum at s = 1/ 2 and hence attains maximum at s = 1/ 2. Therefore, from Theorem 2.5, we have
Example 2.8. Let X = ℝ^{2} with the norm
Then
Proof. It is obvious to check that the norm x = max{x_{2}, λx_{∞}} is absolute, but not normalized, since (1, 0) = (0, 1) = λ. Let us put
Then . ∈ ℕ _{ α } and its corresponding function is
Then
Consider the increasing continuous function . Because g(0) = 1 and , there exists a unique 0 ≤ a ≤ 1 such that g(a) = λ. In fact g(s) is symmetric with respect to s = 1/ 2. Then we have
Obviously, g(s) attains its maximum at s = 1/ 2. Hence, from Theorem 2.4 and Lemma 2.6, we have
Example 2.9. Let X = ℝ^{2} with the norm
Then
Proof. It is obvious to check that the norm is absolute, but not normalized, since (1, 0) = (0, 1) = (1 + λ)^{1/ 2}. Let us put
Therefore, . ∈ ℕ _{ α } and its corresponding function is
Obvious ψ(s) ≤ ψ_{2}(s). Since is symmetric with respect to s = 1/ 2, it suffices to consider for s ∈ [0, 1/ 2]. Note that, for any s ∈ [0, 1/ 2], put . Taking derivative of the function g(s), we have
We always have g'(s) ≥ 0 for 0 ≤ s ≤ 1/ 2. This implies that the function g(s) is increased for 0 ≤ s ≤ 1/2. Therefore, the function attains its maximum at s = 1/2. By Theorem 2.4 and Lemma 2.6, we have
Example 2.10. (Lorentz sequence spaces). Let ω_{1} ≥ ω_{2}> 0, 2 ≤ r < ∞. Twodimensional Lorentz sequence space, i.e. ℝ^{2} with the norm
where is the rearrangement of (z, ω) satisfying , then
Proof. It is obvious that , and the corresponding convex function is given by
Obviously ψ(s) ≤ ψ_{ r }(s) and . It suffices to consider Φ(s) for s ∈ [0, 1/2] since Φ(s) is symmetric with respect to s = 1/2. Note that for s ∈ [0, 1/2]
Some elementary computation shows that u(s)  v(s) = (1(ω_{2}/ω_{1}))s^{r} attains its maximum and v(s) attains its minimum at s = 1/2. Hence,
attains its maximum at s = 1/2 and so does Φ(s). Then by Theorem 2.4 and Lemma 2.6, we have
Example 2.11. Let X be twodimensional Cesà ro space , then
Proof. We first define
for (x, y) ∈ ℝ^{2}. It follows that is isometrically isomorphic to (ℝ^{2}, .) and . is an absolute and normalized norm, and the corresponding convex function is given by
Indeed, defined by is an isometric isomorphism. We prove that ψ(s) ≥ ψ_{2}(s). Note that
Consequently,
Some elementary computation shows that attains its maximum at s = 1/2. Therefore, from Theorem 2.5, we have
3. Constant and uniform normal structure
First, we recall some basic facts about ultrapowers. Let l_{∞}(X) denote the subspace of the product space II_{n∈ℕ}X equipped with the norm (x_{ n } ) := sup_{n∈ℕ}x_{ n }  < ∞. Let be an ultrafilter on ℕ and let
The ultrapower of X, denoted by , is the quotient space equipped with the quotient norm. Write to denote the elements of the ultrapower. Note that if is nontrivial, then X can be embedded into isometrically. We also note that if X is superreflexive, that is , then X has uniform normal structure if and only if has normal structure (see [17]).
Theorem 3.1. Let X be a Banach space with
for some t ∈ (0, 1]. Then X has uniform normal structure.
Proof. Observe that X is uniform nonsquare (see [6]) and then X is superreflexive, it is enough to show that X has normal structure. Suppose that X lacks normal structure, then by Saejung [18, Lemma 2], there exist and satisfying:

(1)
and for all i ≠ j.

(2)
for i = 1, 2, 3.

(3)
.
Let and consider three possible cases.
First, if . In this case, let us put and . It follows that , and
Secondly, if and . In this case, let us put and . It follows that , and
Thirdly, and . In this case, let us put and . It follows that , and
Then, by definition of J_{ X, p } (t) and the fact ,
This is a contradiction and thus the proof is complete.
References
Gao J, Lau KS: On two classes Banach spaces with uniform normal structure. Studia Math 1991, 99: 41–56.
Kato M, Maligranda L, Takahashi Y: On James and Jordanvon Neumann constants and normal structure coefficient of Banach spaces. Studia Math 2001, 144: 275–295. 10.4064/sm14435
Zuo ZF, Cui Y: On some parameters and the fixed point property for multivalued nonexpansive mapping. J Math Sci Adv Appl 2008, 1: 183–199.
Zuo ZZ, Cui Y: A note on the modulus of U convexity and modulus of W*convexity. J Inequal Pure Appl Math 2008,9(4):1–7.
Zuo ZF, Cui Y: Some modulus and normal structure in Banach space. J Inequal Appl 2009, 2009: Article ID 676373.
Zuo ZF, Cui Y: A coefficient related to some geometrical properties of Banach space. J Inequal Appl 2009, 2009: Article ID 934321.
Zuo ZF, Cui Y: The application of generalization modulus of convexity in fixed point theory. J Nat Sci Heilongjiang Univ 2009, 2: 206–210.
Zuo ZF, Cui Y: Some sufficient conditions for fixed points of multivalued nonexpansive mappings. Fixed Point Theory Appl 2009, 2009: Article ID 319804.
LlorensFuster E: The Ptolemy and Zbǎganu constants of normed spaces. Nonlinear Anal 2010, 72: 3984–3993. 10.1016/j.na.2010.01.030
GarciaFalset J: The fixed point property in Banach spaces with NUSproperty. J Math Anal Appl 1997, 215: 532–542. 10.1006/jmaa.1997.5657
Sims B: A class of spaces with weak normal structure. Bull Aust Math Soc 1994, 50: 523–528.
Bonsall FF, Duncan J: Numerical Ranges II. London Mathematical Society Lecture Notes Series, vol. 10. Cambridge University Press, New York; 1973.
Shue JS: On the Ces à ro sequence spaces. Tamkang J Math 1970, 1: 143–150.
Cui Y, Jie L, Pluciennik R: Local uniform nonsquareness in Ces à ro sequence spaces. Comment Math 1997, 27: 47–58.
Cui Y, Hudik H: Some geometric properties related to fixed point theory in Ces à ro spaces. Collect Math 1999,50(3):277–288.
Maligranda L, Petrot N, Suantai S: On the James constant and B convexity of Ces à ro and Ces à roOrlicz sequence spaces. J Math Anal Appl 2007,326(1):312–331. 10.1016/j.jmaa.2006.02.085
Khamsi MA: Uniform smoothness implies supernormal structure property. Nonlinear Anal 1992, 19: 1063–1069. 10.1016/0362546X(92)90124W
Saejung S: Sufficient conditions for uniform normal structure of Banach spaces and their duals. J Math Anal Appl 2007, 330: 597–604. 10.1016/j.jmaa.2006.07.087
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The author wish to express their heartfelt thanks to the referees for their detailed and helpful suggestions for revising the manuscript.
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ZZF designed and performed all the steps of proof in this research and approved the final manuscript.
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Zuo, Z. A geometrical constant and normal normal structure in Banach Spaces. J Inequal Appl 2011, 16 (2011). https://doi.org/10.1186/1029242X201116
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DOI: https://doi.org/10.1186/1029242X201116
Keywords
 Geometrical constant
 Absolute normalized norm
 Lorentz sequence space
 Uniform normal structure