The skew generalized von Neumann-Jordan type constant in Banach spaces

TL;DR

This paper introduces a new skew generalized von Neumann-Jordan type constant for Banach spaces, explores its properties, relations to existing constants, and applications in studying isomorphic spaces and normal structure.

Contribution

It defines a novel constant based on C(X), investigates its properties and relations, and applies it to analyze isomorphic Banach spaces and normal structure conditions.

Findings

Established basic properties of the new constant

Linked the new constant with Banach-Mazur distance and weak orthogonality

Provided a sufficient condition for normal structure in Banach spaces

Abstract

Recently, the von Neumann-Jordan type constants C(X) has defined by Takahashi. A new skew generalized constant Cp({\lambda},\mu,X) based on C(X) constant is given in this paper. First, we will obtain some basic properties of this new constant. Moreover, some relations between this new constant and other constants are investigated. Specially, with the Banach-Mazur distance, we use this new constant to study isomorphic Banach spaces. Ultimately, by leveraging the connection between the newly introduced constant and the weak orthogonality coefficient {\omega}(X), a sufficient condition for normal structure is established.