Some aspects of generalized Zbăganu and James constant in Banach spaces

Qi Liu,Yongjin Li,Muhammad Sarfraz

doi:10.1515/dema-2021-0033

Abstract

Abstract We shall introduce a new geometric constant C Z ( λ , μ , X ) {C}_{Z}\left(\lambda ,\mu ,X) based on a generalization of the parallelogram law, which was proposed by Moslehian and Rassias. First, it is shown that, for a Banach space, C Z ( λ , μ , X ) {C}_{Z}\left(\lambda ,\mu ,X) is equal to 1 if and only if the norm is induced by an inner product. Next, a characterization of uniformly non-square is given, that is, X X has the fixed point property. Also, a sufficient condition which implies weak normal structure is presented. Moreover, a generalized James constant J ( λ , X ) J\left(\lambda ,X) is also introduced. Finally, some basic properties of this new coefficient are presented.

Highlights

A lot of geometric constants have been defined and studied in the literature, which makes it easier for us to deal with some problems in Banach space, because it can describe the geometric properties of space quantitatively, and these geometric constants have mathematical beauty, and there are countless relationships between different geometric constants
It is worth mentioning that geometric constants play a vital role as a tool for solving other problems, such as in the study of Banach-Stone theorem, Bishop-Phelps-Bollobás theorem, and Tingley’s problem
300 Qi Liu et al Inspired by the above theorem, we introduce a new geometric constant CZ(λ, μ, X) in a Banach space X

Summary

Introduction

A lot of geometric constants have been defined and studied in the literature, which makes it easier for us to deal with some problems in Banach space, because it can describe the geometric properties of space quantitatively, and these geometric constants have mathematical beauty, and there are countless relationships between different geometric constants. It is worth mentioning that geometric constants play a vital role as a tool for solving other problems, such as in the study of Banach-Stone theorem, Bishop-Phelps-Bollobás theorem, and Tingley’s problem. These are important research topics in the area of functional analysis and we recommend readers to read the literature [14–16]. If we consider the usual Euclidean space (Rn, ‖⋅‖), the identity ‖x + y‖2 + ‖x − y‖2 = 2‖x‖2 + 2‖y‖2 is called the parallelogram law, and it is well known This identity can be naturally extended to the more general case.

Preliminaries

Weak orthogonality coefficient