Article Search
eISSN 0454-8124
pISSN 1225-6951

### Article

Kyungpook Mathematical Journal 2022; 62(2): 271-287

Published online June 30, 2022

### A New Geometric Constant in Banach Spaces Related to the Isosceles Orthogonality

Zhijian Yang and Yongjin Li*

Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, P. R. China
e-mail : yangzhj55@mail2.sysu.edu.cn and stslyj@mail.sysu.edu.cn

Received: February 25, 2022; Revised: April 8, 2022; Accepted: April 18, 2022

In this paper, starting with the geometric constants that can characterize Hilbert spaces, combined with the isosceles orthogonality of Banach spaces, the orthogonal geometric constant ΩX(α) is defined, and some theorems on the geometric properties of Banach spaces are derived. Firstly, this paper reviews the research progress of orthogonal geometric constants in recent years. Then, this paper explores the basic properties of the new geometric constants and their relationship with conventional geometric constants, and deduces the identity of ΩX(α) and γX(α). Finally, according to the identities, the relationship between these the new orthogonal geometric constant and the geometric properties of Banach Spaces (such as uniformly non-squareness, smoothness, convexity, normal structure, etc.) is studied, and some necessary and sufficient conditions are obtained.

Keywords: Banach spaces, isosceles orthogonality, geometric constants, geometric properties

As we all know, the geometric theory of Banach spaces has been fully developed and synthesizes the properties of concrete spaces such as the classical sequence spaces c0,lp(1p<) and the function space Ca,b. After fifty years of exploration and research, scholars found that some abstract properties of Banach spaces can be quantitatively described by some special constants. At present, there are many papers on geometric constants, but how to use geometric constants to classify Banach spaces is an important problem. For example, Clarkson introduced the module of convexity to be used to characterize uniformly convex spaces [17], and the von-Neumann constant to be used to characterize uniformly non-square spaces and inner product spaces [6]. After, in order to study the normal structure of spaces, James introduced the James constant [13]. After the appearance of these constants, many scholars paid attention to them, and obtained many wonderful properties. Although the study of geometric constants has gone through more than half a century, many new geometric constants constantly appear in our field of vision. Since the 1960s, not only the geometric theory of Banach spaces has been fully developed, but also its research methods have been applied to matrix theory, differential equations and so on.

As the geometric properties of general Hilbert spaces, orthogonal relation has strong geometric intuition. For the general Banach spaces, due to the lack of the definition of inner product, scholars introduced a variety of orthogonality equivalent to the traditional orthogonal relationship in Hilbert spaces. For example, in the real normed space (X,), James [14] defined isosceles orthogonality: xIy if and only if x+y=xy. In 1935, Birkhoff [2] defined Birkhoff orthogonality: xBy if and only if xx+ty. For another example, Roberts [21] defined Robert orthogonality which contains both isosceles and Birkhoff orthogonality: xRy if and only if x+λy=xλy. In addition to the above three orthogonalities, Balestro [4] introduced Pythagorean orthogonality: xPy if and only if x+y2=x2+y2. In Hilbert space, these orthogonalities can be simplified to the orthogonal relation in the traditional sense. However, these orthogonalities are different in general Banach spaces. In order to study the differences between these orthogonalities, a large number of orthogonal geometric constants have been defined and studied [2, 15, 18], including

BR(X)=supα>0x+αyxαyα:x,ySX,xBy

and

BI(X)=supx+yxyx:x,ySX,x,y0,xBy.

The introduction of these orthogonal geometric constants not only enriches the theory of Banach spaces, but also provides important tools for the study of quasi Banach spaces.

Although there are a large number of studies on the differences between these orthogonalities, there are few studies involving Pythagorean orthogonality, especially the differences between isosceles and Pythagorean orthogonality. Therefore, this paper defines a new orthogonal geometric constant with the help of the properties of isosceles orthogonality, as follows:

ΩX(α)=supαx+y2+x+αy2x+y2:xIy,(x,y)(0,0), where 0α<1.

Then, the inequalities between the new constant and the James constant, von-Nuemann constant and the module of convexity are discussed. Finally, the judgment theorems of the geometric properties of Banach spaces are obtained, including uniform non-squareness, uniform convexity, uniform smoothness, strict convexity and uniform normal structure.

In this section, let's recall some concepts of geometric properties of Banach spaces and significant functions.

Definition 2.1. ([5]) Let Xbe Banach space, then the module of convexity is defined as

δX(ε)=inf1x+y2:x,ySX,xy=ε, where ε0,2.

Definition 2.2. ([13]) Let X be Banach space, then the James constant is defined as

J(X)=sup{min{x+y,xy}:x,ySX}.

Definition 2.3. ([6]) Let X be Banach space, then the von-Neumann constant is defined as

CNJ(X)=supx+y2+xy22x2+2y2:x,yX,(x,y)(0,0).

And the modified von-Neumann constant is defined as

CNJ(X)=supx+y2 +xy2 4:x,ySX.

Some famous conclusions about CNJ(X) are listed below:

(i) CNJ(X)J(X) [23];

(ii) 1CNJ(X)2 [16];

(iii) X is a Hilbert space if and only if CNJ(X)=1 [16];

(iv) X is uniformly non-square if and only if CNJ(X)<2 [22].

Definition 2.4. ([25]) Let X be Banach space, then the function γX(t):0,10,4 is defined as

γX(t)=supx+ty2+xty22:x,ySX.

Definition 2.5. ([8]) Let X be Banach space, then the module of smoothness ρX(t) is defined as

ρX(t)=supx+ty+xty21:x,ySX,, where t0,+.

In addition, in order to better characterize the properties of Banach spaces, Zbăganu [7] generalized the constant CNJ(X) in 2001 and introduced the following constant:

Definition 2.6. ([7]) Let X be Banach space, then the Zbăganu constant is defined as

CZ(X)=supx+yxyx2+y2:x,yX,(x,y)(0,0).

Alonso and Martin [24] proved the existence of Banach space X such that CZ(X)<CNJ(X). Now, we review some definitions of the properties of Banach spaces.

Definition 2.7. ([13]) The Banach space X is called uniformly non-square if there exists δ(0,1) such that for any x,ySX, either x+y21δ or xy21δ.

Definition 2.8. ([5]) The Banach spac X is called strictly convex if x=y=1 and xy imply x+y<2.

Definition 2.9. ([5]) The Banach spac X is said to be uniformly convex whenever given 0<ε2, there exists δ>0 such that if x,ySX and xyε, then x+y21δ.

Definition 2.10. ([1]) Let X be Banach space, then diamA=sup{xy:x,yA} is called the diameter of A and r(A)=inf{sup{xy}:yA} is called the Chebyshev radius of A. X is said to have normal structure provided r(A)<diamA for every bounded closed convex subset A of X with diamA>0. X is said to have uniform normal structure if infdiamAr(A)>1 with diamA>0.

Next, list some conclusions about the geometric properties of Banach spaces as follows:

Lemma 2.11. Let X be Banach space, then

(i) If δX(1)>0, then X has normal structure [10].

(ii) X is uniformly non-square if and only if J(X)<2 [11].

(iii) If J(X)<1+52, then X has uniform normal structure [9].

(iv) X is uniformly smooth if and only if limt0+γX(t)1t=0 [25].

(v) If 2γX(t)<1+(1+t)2 for some t0,1, then X has uniform normal structure [25].

(vi) X is strictly convex if and only if δX(2)=1 [25].

(vii) X is uniformly convex if and only if sup{ε0,2:δX(ε)=0}=0 [10].

As we all know, for general normed spaces, the parallelogram rule can describe inner product spaces. In [3], this rule is extended to the following form:

Lemma 2.12. ([3]) Let X be a real normed linear space, then (X,) is an inner product space if and only if for any x,ySX, there exist α,β0 such that

αx+βy2+αxβy2 2(α2+β2),

where ∼ stands for =,≤ or ≥.

In the middle of last century, in order to extend the orthogonal relation of inner product spaces to any Banach spaces, many scholars introduced new orthogonality. For example, as the orthogonality of general Banach spaces, James defined isosceles orthogonality in 1945:

Definition 2.13. ([14]) Let X be Banach space, x,yX, if x+y=xy, then x is called to be isosceles orthogonal to y, denoted as xIy.

As a special case of isosceles orthogonality, Roberts also introduced Roberts orthogonality:

Definition 2.14. ([21]) Let X be Banach space, x,yX, if x+λy=xλy for any λR, then x is called to be Roberts orthogonal to y, denoted as xRy.

In addition to isosceles orthogonality, Birkhoff also defined Birkhoff orthogonality in 1935:

Definition 2.15. ([2]) Let X be Banach space, x,yX, if xx+ty for any tR, then x is called to be Birkhoff orthogonal to y, denoted as xBy.

Moreover, Balestro [4] introduced Pythagorean orthogonality, which is equivalent to orthogonality in the traditional sense in the inner product space.

Definition 2.16. ([4]) Let X be Banach space, x,yX, if x+y2=x2+y2, then x is called to be Pythagorean orthogonal to y, denoted as xPy.

In recent years, based on the geometric constants describing properties of Banach spaces, scholars have defined many new geometric constants with the help of Birkhoff orthogonality and Roberts orthogonality, and explored the properties of Banach spaces [19, 12].

Definition 2.17. ([19]) Let X be Banach space, then the Birkhoff orthogonal geometric constant BR(X) is defined as

BR(X)=supα>0x+αyxαyα:x,ySX,xBy.

From the definition of BR(X), we can see that it can describe the difference between Roberts and Birkhoff orthogonality. Meanwhile Birkhoff orthogonality is homogeneous, it can be thought that BR(X) also measure the difference between Birkhoff and isosceles orthogonalities. In [19], the author proves that BR (X) = 0 and X is Hilbert spaces, and deduces the properties of the corresponding points when BR(X) reaches the supremum.

Recently, in order to explore the difference between Birkhoff orthogonality and isosceles orthogonality, Ji [15] and Mizuguchi [18] have defined two geometric constants, as shown below:

Definition 2.18. ([15]) Let X be Banach space, then the isosceles orthogonal geometric constant D(X) is defined as

D(X)=infinfλRx+λy:x,ySX,xIy.

Definition 2.19. ([18]) Let X be Banach space, then the isosceles orthogonal geometric constant IB(X) is defined as

IB(X)=infinfλRx+λyx:x,yX,x,y0,xIy.

Based on the parallelogram law and isosceles orthogonality, Liu [20] introduced a new geometric constant Ω(X), gave properties of this geometric constant, and used it to characterized the inner product space.

Definition 2.20. ([20]) Let X be Banach space, then the isosceles orthogonal geometric constant Ω(X) is defined as

Ω(X)=sup2x+y2+x+2y2x+y2:x,yX,(x,y)(0,0),xIy.

In this paper, for narrative convenience, we let X be real Banach space with dimX2. The unit ball and the unit sphere of X are denoted by BX and SX, respectively.

### 3. The Isosceles Orthogonal Geometric Constant of Quadratic Form

As is known to all, for the general Banach space X, Pythagorean orthogonality and isosceles orthogonality are not equivalent. But when X is an inner product space, for any two non-zero vectors x, y∈ X and xIy, it is easy to know xy, that is, xPy. Hence x+y2=x2+y2 and

αx+y2+x+αy2=x2+αy2+y+2αx,y+αx2+y2+2αx,y        =x2+α2y2+α2x2+y2        =(1+α2)(x2+y2),

which implie that

αx+y2+x+αy2x+y2=αx+y2+x+αy2x2+y2=1+α2

for any αR. Therefore, in order to explore the difference between Pythagorean orthogonality and isosceles orthogonality, this paper defines the isosceles orthogonal geometric constant of quadratic form, as follows:

Definition 3.21. Let X be Banach space, then the isosceles orthogonal geometric constant of quadratic form is defined as

ΩX(α)=supαx+y2+x+αy2x+y2:xIy,(x,y)(0,0), where 0α<1.

Theorem 3.22. Let X ba Banach space, then 1+α2ΩX(α)2.

Proof. Letting x0=0,y00, then x0Iy0 and

ΩX(α)αx0+y02+x0+αy02x0+y02=1+α2.

Letting x,yX and xIy, then αx+y=1+α2(x+y)1α2(xy) and x+αy=1+α2(x+y)+1α2(xy), thus

αx+y1+α2x+y+1α2xy=x+y

and

x+αy1+α2x+y+1α2xy=x+y,

that is, ΩX(α)2.

Example 3.23. Let l1 be the linear space of all sequences in R such that i=1|xi|< with the norm defined by

x1= i=1|xi|.

Choose x=(1,1,0,), y=(1,1,0,), then xIy and x+y1=αx+y1=x+αy1=2, that is, Ωl1(α)2. Hence Ωl1(α)=2.

Let l be the linear space of all bounded sequences in R with the norm defined by

x=sup1n|xn|.

Choose x=(1,0,),y=(0,1,0,), then xIy and x+y=αx+y=x+αy=1, that is, Ωl(α)2. Hence Ωl(α)=2.

Example 3.24. Let Ca,b be the linear space of all real valued continuous functions on a,b with the norm defined by

x=supta,b|x(t)|.

We choose x0=1ab(tb),y0=1ab(tb)+1SCa,b, then x0Iy0, thus

ΩCa,b(α)αx0+y02+x0+αy02x0+y02    =supta,b 1α ab (tb)+α2+supta,b 1+α ab (tb)α2=2,

which implies that ΩCa,b(α)=2.

Theorem 3.25. Let X be Banach space, then

(i) ΩX(α) is convex and continuous with respect to α0,1.

(ii) ΩX(α) is a non-decreasing function with respect to α0,1.

(iii) ΩX(α)21α is a non-increasing function with respect to α0,1.

Proof. (i) Since 2 is convex and αx+y2+x+αy2=(αx+y,x+αy)22, then ΩX(α) is obviously convex and continuous.

(ii) In order to prove this theorem, we need to extend the definition interval of the constant ΩX(α) to (-1,1), and it is easy to know that ΩX(α)=ΩX(α), α0,1. Setting 0α1<α2<1, then

ΩX(α1)=ΩXα2+α12α2α2+α2α12α2(α2)    α2+α12α2ΩX(α2)+α2α12α2ΩX(α2)=ΩX(α2),

thus ΩX(α) is a non-decreasing function.

(iii) Setting 0α1<α2<1. In order to ensure the continuity of ΩX(α), its supplementary definition is ΩX(1)=2. Then

ΩX(α2)21α2=ΩXα2α11α11+1α2α11α1 α121α2    2α2α11α1+1α2α11α1ΩX(α1)21α2=ΩX(α1)21α1,

thus ΩX(α)21α is a non-increasing function.

Theorem 3.26. Let X be Banach space, then the following conditions are equivalent:

(i) X is Hilbert space.

(ii) ΩX(α)=1+α2 for any α0,1.

(iii) ΩX(α0)=1+α02 for some α00,1.

Proof. Suppose (i) holds, then (ii) clearly holds by the definition of ΩX(α)

Suppose (ii) holds, then (iii) is clearly established.

Suppose (iii) holds, then x+yIxy for any x,ySX. Hence

α0(x+y)+(xy)2+(x+y)+α0(xy)2(x+y)+(xy)21+α02,

that is, (α0+1)x+(1α0)y2+(α0+1)x(1α0)y24(1+α02). Letting a=α0+1,b=1α0, then a,b0 and ax+by2+axby22(a2+b2), which implies that (i) holds.

### 4. Conclusions Related to Other Geometric Constants

In this section, we will study some inequalities for ΩX(α) and some geometric constnats, including the James constant J(X), the von-Neumann constant CNJ(X), the module of convexity δX(ε) and so on. Moreover, these inequalities will help us to discuss the relations between ΩX(α) and some properties of Banach spaces in the next section.

Theorem 4.27. Let X be Banach space, then ΩX(α)=(1+α)22γX1α1+α.

Proof. Letting x,y∈ X and xIy, we set u=x+y2,v=xy2, then

x+αy=(1+α)u+(1α)v,αx+y=(1+α)u(1α)v,

thus u=v and

x+αy2+αx+y2x+y2=(1+α)u+(1α)v2+(1+α)u(1α)v24u2        =(1+α)24u+1α1+α v2+u1α1+α v2u2.

Let x=uu,y=vv, then x,ySX and

u+1α1+αv2+u1α1+αv2u2= x + 1α 1+α y 2+ x 1α 1+α y 2            2γX1α1+α,

that is, ΩX(α)(1+α)22γX1α1+α.

Letting x,ySX, we set u=x+y2, v=xy2, then u+v,uvSX. Since

x+1α1+αy2+x1α1+αy22=u+v+1α1+α(uv)2+u+v1α1+α(uv)22u+v2            =2(1+α)2u+αv2+αu+v2u+v2            2(1+α)2ΩX(α),

then ΩX(α)(1+α)22γX1α1+α.

Example 4.28. Let lp (1<p<) be the linear space of all sequences in R such that i=1|xi|p< with the norm defined by

xp= i=1|xi|p1p.

Choose x0=121p,121p,0,,0,y0=121p,121p,0,,0, then

γlp(t)x0+ty02+x0ty022=(1+t)p +(1t)p 22p,

which implies that

Ωlp(α)(1+α)22 1+1α1+α p +11α1+α p 22p=212p(1+αp)2p.

In particular, if 2p<, then since γlp(t)=(1+t)p+(1t)p 22p [5], we can deduce that

Ωlp(α)=(1+α)22γX1α1+α=212p(1+αp)2p.

Corollary 4.29. Let X be Banach space, then (1α)2 C NJ(X)ΩX(α)(1+α2)CNJ(X).

Proof. Since CNJ(X)γX1α1+α1+1α1+α2=(1+α)2γX1α1+α2+2α2=ΩX(α)1+α2, then ΩX(α)(1+α2)CNJ(X).

Note that x+yIxy for any x,ySX, then

ΩX(α)x+y+α(xy)2+α(x+y)+xy2x+y+xy2    =(1+α)x+(1α)y2+(1+α)x(1α)y24    =(1+α)24(x+ky2+xky2),

where k=1α1+α. Since

x+ky=1+k2(x+y)+1k2(xy)1+k2x+y1k2xy

and

xky=1k2(x+y)+1+k2(xy)1k2x+y1+k2xy,

then we have

x+ky2+xky21+k22(x+y2+xy2)(1k2)x+yxy        k2(x+y2+xy2).

Hence ΩX(α)(1α)24(x+y2+xy2), that is, ΩX(α)(1α)2C NJ(X).

Corollary 4.30. Let X be Banach space, then

(1+α)22J2(X)2α(1+α)J(X)+2α2ΩX(α)1+α24J2(X)+2αJ(X)+1+α2.

Proof. Letting x,ySX, then since x+y=x+1α1+αy+2α1+αyx+1α1+αy+2α1+α and xy=x1α1+αy2α1+αyx1α1+αy+2α1+α, we have

min{x+y,xy}2minx+1α1+αy+2α1+α,x1α1+αy+2α1+α2x+1α1+αy+2α1+α2+x1α1+αy+2α1+α22=x+1α1+αy2+x1α1+αy22+2α1+α(x+1α1+αy+x1α1+αy)+4α2(1+α)2γX1α1+α+4α1+αγX1α1+α+4α2(1+α)2,

which shows that J(X)γX1α1+α+2α1+α=2(1+α)2ΩX(α)+2α1+α. Hence

(1+α)22J2(X)2α(1+α)J(X)+2α2ΩX(α).

In addition, since x+ky=1+k2(x+y)+1k2(xy)1+k2x+y+1k2xy and xky=1k2(x+y)+1+k2(xy)1k2x+y+1+k2xy, then

x+ky2+xky21+k22(x+y2+xy2)+(1k2)x+yxy        1+k22(J2(X)+4)+2(1k2)J(X),

that is, γX1α1+α1+α22(1+α)2J2(X)+4α(1+α)2J(X)+2+2α2(1+α)2. Hence ΩX(α)1+α24J2(X)+2αJ(X)+1+α2.

Corollary 4.31. Let X be Banach space, then

(1α)22(1+ε2δX(ε))2ΩX(α)(1+α2)(1δX(ε)+αε1+α2)2+(1α2)2ε24+4α2,

where ε0,2.

Proof. Since

2γX(t)x+ty2+xty2  1+t22(x+y2+xy2)(1t2)x+yxy  t22(x+y+xy)2,

then γX(t)t2(ρX(1)+1)2, that is,

ΩX(α)=(1+α)22γX1α1+α(1α)22(ρX(1)+1)2.

Note that ρX(1)=supε2δX(ε):0ε2 [17], then

ΩX(α)(1α)221+ε2δX(ε)2.

Since x+y22δX(xy) for any x,ySX, then

x+ty2+xty21+t22(x+y2+xy2)+(1t2)x+yxy1+t22(4(1δX(xy))2+xy2)+2(1t2)(1δX(xy))xy,

which implies that γX(t)(1+t2)(1δX(ε))2+(1t2)(1δX(ε))ε+1+t24ε2. Thus ΩX(α)=(1+α)22γX1α1+α(1+α2)(1δX(ε))2+2αε(1δX(ε))+1+α24ε2.

### 5. Conclusions Related to the Properties of Banach Spaces

In this part, with the help of the inequality of the new constant and the definition of the geometric properties of Banach spaces, the characterization theorems of the new constant for the properties of uniformly non-square, uniformly convex, strictly convex and uniform normal structure of Banach spaces are derived.

Theorem 5.32. Let X be Banach space, then

(i) If X is not uniformly non-square, then ΩX(α)=2 for any α0,1.

(ii) If ΩX(α0)<(1+α0 )24+1 for some α00,1, then X has uniform normal structure.

(iii) If ΩX(α0)<9(1α0 )28 for some α00,1, then X has normal structure.

Proof. (i) Since X is not uniformly non-square, then γX1α1+α=1+1α1+α2=4(1+α)2, that is, ΩX(α)=(1+α)22γX1α1+α=2.

(ii) Since ΩX(α0)<(1+α0 )24+1, then

2γX1α0 1+α0 =4(1+α0 )2ΩX(α0)<1+4(1+α0 )2=1+1+1α01+α02,

which shows that X uniform normal structure.

(iii) Since ΩX(α0)<9(1α0 )28, then (1α0)221+ε2 δX(ε)2<9(1α0)28, that is, δX(ε)>ε12, thus δX(1)>0. Hence X has normal structure.

Example 5.33. Let lplq (1qp<) be R2 with the norm defined by

(x1,x2)=(x1 ,x2 )p  , x1 x2 0(x1 ,x2 )q  , x1 x2 <0.

Let ll1 be R2 with the norm defined by

(x1,x2)=(x1 ,x2 )  , x1 x2 0(x1 ,x2 )1  , x1 x2 <0.

We choose x0=121p,121p,y0=121q,121q, then

γlplq(t)x0+ty02+x0ty022=22p1+t21p1q p+1t21p1q p2p,

which implies that

Ωlplq(α)=(1+α)22γlplq1α1+α   212p 1+ 2 1 p1 q+ 1 2 1 p1 qα p + 1 2 1 p1 q+ 1+ 2 1 p1 qα p 2p.

In particular, since γl2l1(t)=1+t+t2 and γll1(t)=12(1+(1+t)2) [5], then

Ωl2l1(α)=(1+α)22γl2l11α1+α=3+α22

and

Ωll1(α)=(1+α)22γll11α1+α=(1+α)24+1.

Thus Ωl2l1(α),Ωll1(α)<2 for any α0,1, which implies that l2l1, ll1 both are uniformly non-square.

Corollary 5.34. Let X be a finite dimensional Banach space, if ΩX(α0)=2 for some α00,1, then X is not uniformly non-square.

Proof. Since ΩX(α0)=2, then there exist xnSX,ynBX such that xnIyn and

limnxn+α0yn2+α0xn+yn2xn+yn2=2.

Since X is finite dimensional, then there exist x0,y0BX such that x0Iy0 and

limkxnk=x0,limkynk=y0.

Note that xn+α0ynxn+yn, α0xn+ynxn+yn and

xn+yn2+xn+yn2xn+yn22,

then x0+α0y0=x0+y0 and α0x0+y0=x0+y0.

Since x0+α0y0(1α0)x0+α0x0+y0, then x0+y0x0. In addition, we also can prove x0+y0y0, then

max{x0+y0,x0y0}=x0+y0min{x0,y0}1<1+δ

for any δ(0,1), which implies that X is not uniformly non-square.

Corollary 5.35. Let X be Banach space and ΩX(0)>1, then X is not uniformly convex. In particular, if ΩX(α)>1+α2 for any α0,1, then X is not strictly convex.

Proof. If ΩX(0)>1, then there exists ε00,2 such that ΩX(0)1+ε024.

Since ΩX(α)(1+α2)1δX(ε)+αε1+α2 2+(1α2 )2ε24+4α2, then

1+ε024(1δX(ε0))2+ε024,

that is, δX(ε0)=0. Thus sup{ε0,2:δX(ε)=0}ε0>0, that is, X is not uniformly convex.

In particular, if ΩX(α)>1+α2, we assume that X is strictly convex, then δX(2)=1. Hence

1+α2<(1+α2)11+2α 1+α22+(1α2 )21+α2=1+α2,

this is contraditory, then X is not strictly convex.

Theorem 5.36. Let X be Banach space, then X is uniformly smooth if and only if

limα11+αΩX(α)1α2=12.

Proof. Letting t=1α1+α, then α=1t1+t and α1t0+. Thus we can get

1+αΩX(α)1α2=1+α(1+α)22γX1α1+α1α2      =1+1t1+t2(1+t)2γX(t)11t1+t 2=1+tγX(t)2t.

Therefore limt0+1+tγX(t)2t=12 if and only if limt0+1γX(t)t=0. That is,

limt0+1+tγX(t)2t=12

if and only if X is uniformly smooth.

No data were used to support this study.

The author(s) declare(s) that there is no conflict of interest regarding the publication of this paper.

1. M. Brodskii and D. Milman, On the center of a convex set, Doklady Akad. Nauk SSSR (N.S.), 59(1948), 837-840.
2. G. Birkhoff, Orthogonality in linear metric spaces, Duke Math. J., 1(1935), 169-172.
3. C. Bentez and M. del Rio, Characterization of inner product spaces through rectangle and square inequalities, Rev. Roumaine Math. Pures Appl., 29(1984), 543-546.
4. V. Balestro, Angles in normed spaces, Aequat. Math., 91(2017), 201-236.
5. J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc., 40(1936), 396-414.
6. J. A. Clarkson, The von Neumann-Jordan constant for the Lebesgue space, Ann. of Math., 38(1937), 114-115.
7. Y. Cui, Some properties concerning Milman's moduli, J. Math. Anal. Appl., 329(2007), 1260-1272.
8. M. M. Day, Uniform convexity in factor and conjugate spaces, Ann. of Math., 45(1944), 375-385.
9. S. Dhompongsa, A. Kaewkhao and S. Tasena, On a generalized James constant, J. Math. Anal. Appl., 285(2003), 419-435.
10. K. Goebel, Convexity of balls and fixed point theorems for mapping with nonexpansive square, Compositio Math., 22(1970), 269-274.
11. J. Gao and K. S. Lau, On the geometry of spheres in normed linear spaces, J. Austral. Math. Soc. Ser. A, 48(1990), 101-112.
12. C. Hao and S. Wu, Homogeneity of isosceles orthogonality and related inequalities, J. Inequal. Appl., 84(2011).
13. R. C. James, Uniformly non-square Banach spaces, Ann. of Math., 80(1964), 542-550.
14. R. C. James, Orthogonality in normed linear spaces, Duke Math. J., 12(1945), 291-302.
15. D. Ji and S. Wu, Quantitative characterization of the difference between Birkhoff orthogonality and isosceles orthogonality, J. Math. Anal. Appl., 323(2006), 1-7.
16. P. Jordan and J. Von Neumann, On inner products in linear metric spaces, . Ann. Math. J., 36(1935), 719-723.
17. J. Lindenstrauss, On the modulus of smoothness and divergent series in Banach spaces, Michigan Math. J., 10(1963), 241-252.
18. H. Mizuguchi, The constants to measure the differences between Birkhoff and isosceles orthogonalities, Filomat, 30(2015), 2761-2770.
19. P. L. Papini and S. Wu, Measurements of differences between orthogonality types, J. Math. Anal. Appl., 397(2013), 285-291.
20. L. Qi and Y. Zhijian, New geometric constants of isosceles orthogonal type, e-print arXiv, (), 2111.08392.
21. B. D. Roberts, On the geometry of abstract vector spaces, Tohoku Math, 39(1934), 42-59.
22. Y. Takahashi and M. Kato, Von Neumann-Jordan constant and uniformly nonsquare Banach spaces, Nihonkai Math. J., 9(1998), 155-169.
23. Y. Takahashi and M. Kato, A simple inequality for the von Neumann-Jordan and James constants of a Banach space, J. Math. Anal. Appl., 359(2009), 602-609.
24. F. Wang and B. Pang, Some inequalities concering the James constant in Banach spaces, J. Math. Anal. Appl., 353(2009), 305-310.
25. C. Yang and F. Wang, On estimates of the generalized Jordan-von Neumann constant of Banach spaces, JIPAM. J. Inequal. Pure Appl. Math., 7(2006), 1-5.