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A new geometry property and two new moduli are introduced in Banach space. First, the concept of local uniform Kadec-Klee property (

Let

It is well known that the condition equivalent to near uniform convexity (

The functions

Recall also that a function

J. Banas’ proved that if

The modulus of

There are many recent papers concerning the Kadec-Klee property, such as Kadec¨CKlee property and fixed points and Dual Kadec-Klee property and fixed points studied by Jean Saint Raymond (see [

Since Banach space is more extensive than Hilbert space, it is quite difficult to describe its geometry structure. An effective method is to introduce new geometric properties for Banach space and to define an appropriate function, usually called a modulus or a geometric constant. Because the range of values of these geometric constants directly determines the existence of some geometric properties; therefore, many scholars are interested to calculate the modules and constants of some specific spaces. The starting point of the present paper is the observation that the

Before starting with our results, we need to recall some notions and a lemma in [

We say that a Banach space

We say that a Banach space

We say that a Banach space

We say that a Banach space

i.e.,

Let

In this paper, we take Kutazrova and Bor-Luh Lin’s approach to localize the

We begin this section by formulating some definitions.

A Banach space

i.e.,

Let

For every

If a Banach space

We prove the contrapositive. Suppose

i.e.,

The following conclusion follows from the definitions of

If Banach space

It follows from previous Corollaries, we conclude the following Corollary.

For every Banach space

We are now ready to prove the main theorems of this paper.

If a Banach space

Suppose that

Since

Case (i) if

Case (ii) if

It is obvious that

For another facts, for

i.e.,

Thus,

For every Banach space

Fix

By the corollary of Hahn-Banach theorem, there exists

It is obvious that the set

Since

Then, there exists

Since

Consequently,

Thus,

For Banach space

For every

Then, there exists a subsequence

Since

If

Fix

And consequently,

From (

Thus,

In this paper, we introduce a new geometric property

No data were used to support this study.

The authors declare that there is no conflict of interest regarding the publication of this paper.

The authors are grateful to the referee for comments which improved the paper. This paper is supported by “The National Science Foundation of China” (11871181); “The Science Research Project of Inner Mongolia Autonomous Region” (NJZY18253); “The Science Research Project of Ordos Institution of Applied Technology” (KYYB2017014).