### Article

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

**Published online** June 30, 2022

Copyright © Kyungpook Mathematical Journal.

### 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

### Abstract

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

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

### 1. Introduction

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

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

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.

### 2. Notations and Preliminaries

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

**Definition 2.1.** ([5]) Let

**Definition 2.2.** ([13]) Let

**Definition 2.3.** ([6]) Let

And the modified von-Neumann constant is defined as

Some famous conclusions about

(i)

(ii)

(iii)

(iv)

**Definition 2.4.** ([25]) Let

**Definition 2.5.** ([8]) Let _{X}(t)

In addition, in order to better characterize the properties of Banach spaces, Zbăganu [7] generalized the constant

**Definition 2.6.** ([7]) Let

Alonso and Martin [24] proved the existence of Banach space

**Definition 2.7.** ([13]) The Banach space

**Definition 2.8.** ([5]) The Banach spac

**Definition 2.9.** ([5]) The Banach spac

**Definition 2.10.** ([1]) Let

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

**Lemma 2.11.** Let

(i) If

(ii)

(iii) If

(iv)

(v) If

(vi)

(vii)

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

where ∼ stands for

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

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

**Definition 2.14.** ([21]) Let

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

**Definition 2.15.** ([2]) Let

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

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

From the definition of

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

**Definition 2.19.** ([18]) Let

Based on the parallelogram law and isosceles orthogonality, Liu [20] introduced a new geometric constant

**Definition 2.20.** ([20]) Let

In this paper, for narrative convenience, we let _{X}_{X}

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

As is known to all, for the general Banach space

which implie that

for any

**Definition 3.21.** Let

**Theorem 3.22.** Let

Letting

and

that is,

**Example 3.23.** Let _{1}

Choose

Let

Choose

**Example 3.24.** Let

We choose

which implies that

**Theorem 3.25.** Let

(i)

(ii)

(iii)

(ii) In order to prove this theorem, we need to extend the definition interval of the constant

thus

(iii) Setting

thus

**Theorem 3.26.** Let

(i)

(ii)

(iii)

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

Suppose (iii) holds, then

that is,

### 4. Conclusions Related to Other Geometric Constants

In this section, we will study some inequalities for

**Theorem 4.27.** Let

thus

Let

that is,

Letting

then

**Example 4.28.** Let

Choose

which implies that

In particular, if

**Corollary 4.29.** Let

Note that

where

and

then we have

Hence

**Corollary 4.30.** Let

which shows that

In addition, since

that is,

**Corollary 4.31.** Let

where

then

Note that

Since

which implies that

### 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

(i) If

(ii) If

(iii) If

(ii) Since

which shows that

(iii) Since

**Example 5.33.** Let ^{2}

Let ^{2}

We choose

which implies that

In particular, since

and

Thus

**Corollary 5.34.** Let

Since

Note that

then

Since

for any

**Corollary 5.35.** Let

Since

that is,

In particular, if

this is contraditory, then

**Theorem 5.36.** Let

Therefore

if and only if

### Data Availability

No data were used to support this study.

### Conflicts of Interest

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

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