#### Abstract

In this paper, we further studied properties of the modulus of -dimensional -convexity and the modulus of -dimensional -flatness when *n* = 1 (2-dimensional character) and *n* = 2 (3-dimensional character). The new properties of these moduli are investigated, and the relationships between these moduli and other geometric parameters of Banach spaces are studied. Some results on fixed point theory for nonexpansive mappings and normal structure in Banach spaces are obtained.

#### 1. Introduction

Let be a real Banach space with the dual space . Let and be the unit ball and the unit sphere of , respectively. Let be the dual space of . Let denote the set of norm 1 supporting functionals of .

Brodskiĭ and Mil’man [1] introduced the following geometric concepts in 1948.

*Definition 1. *Let be a Banach space. A nonempty bounded and convex subset of is said to have normal structure if, for every convex subset of that contains more than one point, there is a point such thatA Banach space is said to have(a)Normal structure if every bounded convex subset of has normal structure.(b)Weak normal structure if every weakly compact convex set of has normal structure.(c)Uniform normal structure if there exists such that, for every bounded closed convex subset of that contains more than one point, there is a point such that

*Remark 1. *The following facts are known:(a)Uniform normal structure normal structure weak normal structure(b)For a reflexive spaces, normal structure weak normal structureKirk [2] proved that if a Banach space has weak normal structure, then it has weak fixed point property, that is, every nonexpansive mapping from a weakly compact and convex subset of to itself has a fixed point.

Let be the set of all natural numbers and .

For two sets of vectors and , the matrix,is denoted by [3].

Gao and Saejung [3] introduced the concept of volume by the convex hull of in ofwhere the supremum is taken over all , where .

*Definition 2. *(see [3]). Let be the upper bound of all -dimensional volume in .

The following result was proved [3].

Proposition 1. *For a Banach space with , we have .*

Gao and Saejung introduced the concept of the modulus of -dimensional -convexity of as follows.

*Definition 3. *(see [3]). Let be a Banach space. Definewhere is the modulus of -dimensional -convexity of .

The following results were proved in [3] too.

Theorem 1. (a)* is an increasing and continuous function in for any .*(b)*If is a Banach space with for some , then is super-reflexive.*(c)*If is a Banach space with for some , then has uniform normal structure**The following results were proved in [4].*

Theorem 2. *If is a Banach space with for some and n is even, then is reflexive.*

Let , and we have the following.

Corollary 1. *If is a Banach space with , then is reflexive.*

The following results were also proved in [4].

Theorem 3. *If is a Banach space with and , then has uniform normal structure.*

Gao and Saejung also introduced the concept of the modulus of -dimensional -flatness as follows.

*Definition 4. *(see [5]). Let be a Banach space. Definewhere is the modulus of -dimensional -flatness of .

*Remark 2. *(a) The name of the modulus of -flatness is defined by comparing with the name of the modulus of -convexity of . (b) in .

The following results were proved in [5] too.

Theorem 4. (a)* is an increasing and continuous function in for any .*(b)*If is a Banach space with , or , then is reflexive.*(c)*If is a Banach space with , where , then is super-reflexive.*(d)*Suppose that is a Banach space satisfying one of the following conditions:(i) , for some with (ii), for or *

*Then, has uniform normal structure.*

Recently, Gabeleh introduced concepts of pointwise cyclic relatively nonexpansive mapping and weak proximal normal structure for an extension of geometric property of normal structure. Some interesting results are obtained there [6].

In this paper, we studied further properties of the modulus of -dimensional -convexity and the modulus of -dimensional -flatness when *n* = 1 (2-dimensional character) and *n* = 2 (3-dimensional character). The new properties of these moduli are investigated and the relationships between these moduli and other geometric parameters of Banach spaces are studied. Some results on fixed point theory for nonexpansive mappings and normal structure in Banach spaces are obtained.

#### 2. Main Results

Lemma 1 (Bishop-Phelps-Bollobás, see [7]). *Let be a Banach space, and let . Given and with , then there exist and such that and .*

Since is a continuous function in norm for and norm for , Lemma 1 can be stated as follows.

Lemma 2. *Let be a Banach space, and let . Given and with , then there exist and such that and .*

Theorem 5 (see [8]). *Let be a Banach space. Then, is not reflexive if and only if, for any , there are a sequence and a sequence such that*(a)* whenever *(b)* whenever *

Theorem 6. *If is a Banach space with , or , then is reflexive.*

*Proof. *Suppose that is not reflexive. Let be given.

Let , and . Then, we haveWe also haveSo,Since can be arbitrarily close to 1, by using Lemma 2 and the definition of and , we have if , or , then is reflexive.

Theorem 7. *If is a Banach space with , or , then is reflexive.*

*Proof. *Suppose that is not reflexive. Let be given.

Let , and .

Then, we haveWe also haveWe haveSince can be arbitrarily close to 1, by using Theorem 2.2 and the definition of and , we have if , or , then is reflexive.

The following result refers to a Banach space with weak sequentially compact unit ball of the dual. Notice that this property is satisfied by reflexive or separable Banach spaces and by those that admit an equivalent smooth norm.

Lemma 3 (see [9]). *If is a Banach space with weak sequentially compact and fails to have weak normal structure, then, for any , there are sequence and sequence such that*(a)*, whenever *(b)*, whenever *(c)*, whenever *(d)*, whenever *

Theorem 8. *If is a Banach space with , or , then has weak normal structure.*

*Proof. *Suppose that does not have weak normal structure. Let be given. Since , we have .

Let ; and . Then, we havewhere *a*, *b*, and *c* are constants and is arbitrarily small.

We also havewhere *d* is a constant.

So,By using Theorem 2.2 and the definition of and , we have if , or , then has weak normal structure.

Theorem 9. *If is a Banach space with , or , then has weak normal structure.*

*Proof. *Suppose that does not have weak normal structure. Let be given.

Let ; and . Then, similar to the proof of Theorem 8, we havewhere *a*, *b*, *c*, *d*, *e*, *f*, and are constants and is arbitrarily small.

We also haveand we haveBy using Theorem 2.2 and the definition of and , we have if , or , then has weak normal structure.

*Definition 5. *(see [10, 11]). Let and be Banach spaces. We say that is finitely representable in if, for any and any finite dimensional subspace , there is an isomorphism such that, for any , .

We say that is super-reflexive if any space which is finitely representable in is reflexive.

Theorem 10. *Suppose that is a Banach space satisfying one of the following conditions:*(a)*(b)**(c)**(d)**Then, is super-reflexive.*

*Proof. *The proof of this theorem follows from the fact that if a Banach space is finitely representable in , then and , for any and for any .

We consider the uniform normal structure. To discuss this result, let us recall the concept of the “ultra”-technique.

Let be a filter of an index set , and let be a subset in a Hausdorff topological space , is said to converge to with respect to , denoted by , if, for each neighborhood of , . A filter on is called an ultrafilter if it is maximal with respect to the ordering of the set inclusion. An ultrafilter is called trivial if it is of the form for some . We will use the fact that if is an ultrafilter, then(i)For any , either or (ii)If has a cluster point , then exists and equals to Let be a family of Banach spaces and let denote the subspace of the product space equipped with the norm .

*Definition 6. *(see [12, 13]). Let be an ultrafilter on and let . The ultraproduct of is the quotient space equipped with the quotient norm.

We will use to denote the element of the ultraproduct. It follows from remark (ii) above and the definition of quotient norm thatIn the following, we will restrict our index set to be , the set of natural numbers, and let , for some Banach space . For an ultrafilter on , we use to denote the ultraproduct. Note that if is nontrivial, then can be embedded into isometrically.

Lemma 4 (see [13]). *Suppose that is an ultrafilter on and is a Banach space. Then, if and only if is super-reflexive; in this case, the mapping defined bywhich is the canonical isometric isomorphism from onto .*

Theorem 11. *Let be a super-reflexive Banach space. Then, for any nontrivial ultrafilter on and for all and , we have , and .*

*Proof. *The proof is the same as the proof of Theorem 2.17 in [3].

Lemma 5 (see [14]). *If is a super-reflexive Banach space, then has uniform normal structure if and only if has normal structure.*

Theorem 12. *Suppose that is a Banach space satisfying one of the following conditions:*(a)*(b)** and , or and , or and , or and *(c)*(d)** and , or and , or and , or and **Then, has uniform normal structure.*

*Proof. *The results (a) and (c) follow directly from Theorem 8, Theorems 10 and 11, and Lemma 5. The results (b) and (d) follow directly from Theorems 9, 10, and 11 and Lemma 5.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.