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(1≤p<∞) 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: x⊥Iy if and only if ‖x+y‖=‖x−y‖. In 1935, Birkhoff [2] defined Birkhoff orthogonality: x⊥By if and only if ‖x‖≤‖x+ty‖. For another example, Roberts [21] defined Robert orthogonality which contains both isosceles and Birkhoff orthogonality: x⊥Ry if and only if ‖x+λy‖=‖x−λy‖. In addition to the above three orthogonalities, Balestro [4] introduced Pythagorean orthogonality: x⊥Py if and only if ‖x+y‖2=‖x‖2+‖y‖2. 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

and

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:

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

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

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

And the modified von-Neumann constant is defined as

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

(i) CNJ(X)≤J(X) [23];

(ii) 1≤CNJ(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):0,1→0,4 is defined as

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

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

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,y∈SX, either ‖x+y‖2≤1−δ or ‖x−y‖2≤1−δ.

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

Definition 2.9. ([5]) The Banach spac X is said to be uniformly convex whenever given 0<ε≤2, there exists δ>0 such that if x,y∈SX and ‖x−y‖≥ε, then x+y2≤1−δ.

Definition 2.10. ([1]) Let X be Banach space, then diamA=sup{‖x−y‖:x,y∈A} is called the diameter of A and r(A)=inf{sup{‖x−y‖}:y∈A} 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 limt→0+γX(t)−1t=0 [25].

(v) If 2γX(t)<1+(1+t)2 for some t∈0,1, 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{ε∈0,2: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,y∈SX, there exist α,β≠0 such that

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,y∈X, if ‖x+y‖=‖x−y‖, then x is called to be isosceles orthogonal to y, denoted as x⊥Iy.

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

Definition 2.14. ([21]) Let X be Banach space, x,y∈X, if ‖x+λy‖=‖x−λy‖ for any λ∈R, then x is called to be Roberts orthogonal to y, denoted as x⊥Ry.

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

Definition 2.15. ([2]) Let X be Banach space, x,y∈X, if ‖x‖≤‖x+ty‖ for any t∈R, then x is called to be Birkhoff orthogonal to y, denoted as x⊥By.

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,y∈X, if ‖x+y‖2=‖x‖2+‖y‖2, then x is called to be Pythagorean orthogonal to y, denoted as x⊥Py.

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

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

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

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

In this paper, for narrative convenience, we let X be real Banach space with dimX≥2. The unit ball and the unit sphere of X are denoted by B_X and S_X, respectively.

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 x⊥Iy, it is easy to know x⊥y, that is, x⊥Py. Hence ‖x+y‖2=‖x‖2+‖y‖2 and

which implie that

for any α∈R. 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

Theorem 3.22. Let X ba Banach space, then 1+α2≤ΩX(α)≤2.

Proof. Letting x0=0,y0≠0, then x0⊥Iy0 and

Letting x,y∈X and x⊥Iy, then αx+y=1+α2⋅(x+y)−1−α2⋅(x−y) and x+αy=1+α2⋅(x+y)+1−α2⋅(x−y), thus

and

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

Choose x=(1,1,0,⋯), y=(1,−1,0,⋯), then x⊥Iy and ‖x+y‖1=‖αx+y‖1=‖x+αy‖1=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

Choose x=(1,0,⋯),y=(0,−1,0,⋯), then x⊥Iy 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

We choose x0=1a−b(t−b),y0=−1a−b(t−b)+1∈SCa,b, then x0⊥Iy0, thus

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 α∈0,1.

(ii) ΩX(α) is a non-decreasing function with respect to α∈0,1.

(iii) ΩX(α)−21−α is a non-increasing function with respect to α∈0,1.

Proof. (i) Since ‖⋅‖2 is convex and ‖αx+y‖2+‖x+αy‖2=‖(‖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), α∈0,1. Setting 0≤α1<α2<1, then

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

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

thus ΩX(α)−21−α 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 α∈0,1.

(iii) ΩX(α0)=1+α02 for some α0∈0,1.

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+y⊥Ix−y for any x,y∈SX. Hence

that is, ‖(α0+1)x+(1−α0)y‖2+‖(α0+1)x−(1−α0)y‖2≤4(1+α02). Letting a=α0+1,b=1−α0, then a,b≠0 and ‖ax+by‖2+‖ax−by‖2≤2(a2+b2), which implies that (i) holds.

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)22γX1−α1+α.

Proof. Letting x,y∈ X and x⊥Iy, we set u=x+y2,v=x−y2, then

thus ‖u‖=‖v‖ and

Let x′=u‖u‖,y′=v‖v‖, then x′,y′∈SX and

that is, ΩX(α)≤(1+α)22γX1−α1+α.

Letting x,y∈SX, we set u=x+y2, v=x−y2, then u+v,u−v∈SX. Since

then ΩX(α)≥(1+α)22γX1−α1+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

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

which implies that

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

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

Proof. Since CNJ(X)≥γX1−α1+α1+1−α1+α2=(1+α)2γX1−α1+α2+2α2=ΩX(α)1+α2, then ΩX(α)≤(1+α2)CNJ(X).

Note that x+y⊥Ix−y for any x,y∈SX, then

where k=1−α1+α. Since

and

then we have

Hence ΩX(α)≥(1−α)24(‖x+y‖2+‖x−y‖2), that is, ΩX(α)≥(1−α)2C′ NJ(X).

Corollary 4.30. Let X be Banach space, then

Proof. Letting x,y∈SX, then since ‖x+y‖=x+1−α1+αy+2α1+αy≤x+1−α1+αy+2α1+α and ‖x−y‖=x−1−α1+αy−2α1+αy≤x−1−α1+αy+2α1+α, we have

which shows that J(X)≤γX1−α1+α+2α1+α=2(1+α)2ΩX(α)+2α1+α. Hence

In addition, since ‖x+ky‖=1+k2(x+y)+1−k2(x−y)≤1+k2‖x+y‖+1−k2‖x−y‖ and ‖x−ky‖=1−k2(x+y)+1+k2(x−y)≤1−k2‖x+y‖+1+k2‖x−y‖, then

that is, γX1−α1+α≤1+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

where ε∈0,2.

Proof. Since

then γX(t)≥t2(ρX(1)+1)2, that is,

Note that ρX(1)=supε2−δX(ε):0≤ε≤2 [17], then

Since ‖x+y‖≤2−2δX(‖x−y‖) for any x,y∈SX, then

which implies that γX(t)≤(1+t2)(1−δX(ε))2+(1−t2)(1−δX(ε))ε+1+t24ε2. Thus ΩX(α)=(1+α)22γX1−α1+α≤(1+α2)(1−δX(ε))2+2αε(1−δX(ε))+1+α24ε2.

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 α∈0,1.

(ii) If ΩX(α0)<(1+α0 )24+1 for some α0∈0,1, then X has uniform normal structure.

(iii) If ΩX(α0)<9(1−α0 )28 for some α0∈0,1, then X has normal structure.

Proof. (i) Since X is not uniformly non-square, then γX1−α1+α=1+1−α1+α2=4(1+α)2, that is, ΩX(α)=(1+α)22γX1−α1+α=2.

(ii) Since ΩX(α0)<(1+α0 )24+1, then

which shows that X uniform normal structure.

(iii) Since ΩX(α0)<9(1−α0 )28, then (1−α0)221+ε2 −δX(ε)2<9(1−α0)28, that is, δX(ε)>ε−12, thus δX(1)>0. Hence X has normal structure.