Generalized geometric constants related to Birkhoff orthogonality in Banach spaces

TL;DR

This paper introduces two new geometric constants in Banach spaces based on Birkhoff orthogonality, explores their properties, and relates them to uniform non-squareness and convexity, with applications in the geometry of Banach spaces.

Contribution

It defines and investigates two generalized geometric constants related to Birkhoff orthogonality, linking them to key Banach space properties and applications.

Findings

Established bounds for the new constants

Characterized uniform non-squareness using these constants

Connected one constant to the modulus of convexity

Abstract

In this paper, based on Birkhoff orthogonality, we introduce two geometric constants $\boldsymbol{A}_{\boldsymbol{t}}^{\boldsymbol{B}}(\boldsymbol{X})$ and $\boldsymbol{D}_{\boldsymbol{t}}^{\boldsymbol{B}}(\boldsymbol{X})$ in Banach spaces, which generalize the skew geometric constants related to Birkhoff orthogonality. We systematically investigate the basic properties of the two constants, including their upper and lower bounds, and establish the equivalent characterizations for Banach spaces being uniformly non-square. Additionally, we explore the relationship between $\boldsymbol{D}_{\boldsymbol{t}}^{\boldsymbol{B}}(\boldsymbol{X})$ and the modulus of convexity $\boldsymbol{\delta}_{\boldsymbol{X}}(\boldsymbol{\varepsilon})$. Finally, we explore several applications of the two newly proposed geometric constants.