# A geometrical constant and normal normal structure in Banach Spaces

- Zhanfei Zuo
^{1}Email author

**2011**:16

https://doi.org/10.1186/1029-242X-2011-16

© Zuo; licensee Springer. 2011

**Received: **1 March 2011

**Accepted: **23 June 2011

**Published: **23 June 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.

## Keywords

## 1. Introduction and preliminaries

*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 non-empty bounded closed convex subset of a Banach space

*X*. A mapping

*T*:

*C*→

*C*is said to be non-expansive provided the inequality

holds for every *x*, *y* ∈ *C*. A Banach space *X* is said to have the fixed point property if every non-expansive mapping *T* : *C* → *C* has a fixed point, where *C* is a non-empty bounded closed convex subset of a Banach space *X*.

*X*is called uniformly non-square 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

*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 Neumann-Jordan 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 non-squareness 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*).

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 García-Falset 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 one-to-one 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
} .

*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 non-*l*_{
r
} norms on ℝ^{2}. One of the properties of these norms is stated in the following result.

The Ces*à* 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.

## 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 2.2**. Let

*X*be the space

*l*

_{ r }or

*L*

_{ r }[0, 1] with dim

*X*≥ 2 (see [6])

- (1)
Let 1

*< r*≤ 2 and 1*/r*+ 1*/r*' = 1. Then for all*t >*0

*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

*r*≥

*p*and the function attains its maximum at

*s*= 1

*/*2, i.e., . From Proposition 2.2, implies that

*x*= (

*a, a*)

*, y*= (

*a*, -

*a*), where . Hence ||

*x*||

_{ ψ }= ||

*y*||

_{ ψ }= 1, and

**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

*p < r*' and the function attains its maximum at

*s*= 1

*/*2, i.e., . From Proposition 2.2, it implies that

**Lemma 2.6**(see [6]). Let || · || and |.| be two equivalent norms on a Banach space. If

*a*|.| ≤ || · || ≤

*b*|.| (

*b*≥

*a >*0), then

**Proof**. It is very easy to check that ||

*x*|| = max{||

*x*||

_{2}

*, λ*||

*x*||

_{1}} ∈ ℕ

_{ α }and its corresponding function is

*ψ*

_{2}(

*s*) attains minimum at

*s*= 1

*/*2 and hence attains maximum at

*s*= 1

*/*2. Therefore, from Theorem 2.5, we have

**Proof**. It is obvious to check that the norm ||

*x*|| = max{||

*x*||

_{2}

*, λ*||

*x*||

_{∞}} is absolute, but not normalized, since ||(1, 0)|| = ||(0, 1)|| =

*λ*. Let us put

*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

**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

*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

*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 <*∞. Two-dimensional Lorentz sequence space, i.e. ℝ

^{2}with the norm

*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]

*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,

## 3. Constant and uniform normal structure

*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 non-trivial, then *X* can be embedded into
isometrically. We also note that if *X* is super-reflexive, that is
, then *X* has uniform normal structure if and only if
has normal structure (see [17]).

for some *t* ∈ (0, 1]. Then *X* has uniform normal structure.

**Proof**. Observe that

*X*is uniform non-square (see [6]) and then

*X*is super-reflexive, 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)
- (2)
- (3)

Let and consider three possible cases.

This is a contradiction and thus the proof is complete.

## Declarations

### Acknowledgements

The author wish to express their heartfelt thanks to the referees for their detailed and helpful suggestions for revising the manuscript.

## Authors’ Affiliations

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