On some generalized geometric constants with two parameters in Banach spaces

TL;DR

This paper introduces two generalized geometric constants in Banach spaces, extending the TX constant, and uses them to characterize Hilbert and non-square spaces through their properties and relationships.

Contribution

The paper develops two new parameterized geometric constants in Banach spaces and demonstrates their utility in characterizing Hilbert and uniformly non-square spaces.

Findings

Derived bounds for the new constants.

Characterized Hilbert spaces using these constants.

Linked the constants to existing Banach space constants.

Abstract

In this paper, we build upon the TX constant that was introduced by Alonso and Llorens-Fuster in 2008. Through the incorporation of suitable parameters, we have successfully generalized the aforementioned constant into two novel forms of geometric constants, which are denoted as T1({\lambda},\mu,X ) and T2(\k{appa},{\tau},X ). First, we obtained some basic properties of these two constants, such as the upper and lower bounds. Next, these two constants served as the basis for our characterization of Hilbert spaces. More significantly, our findings reveal that these two constants exhibit a profound and intricate interrelation with other well-known constants in Banach spaces. Finally, we characterized uniformly non-square spaces by means of these two constants.