On some generalized geometric constants with two parameters in Banach spaces

TL;DR

The paper generalizes the TX constant to two-parameter families in real Banach spaces by introducing $\mathcal{T}_1(\kappa,\tau,\mathcal{X})$ and $\mathcal{T}_2(\kappa,\tau,\mathcal{X})$. It establishes fundamental bounds, derives exact expressions in spaces such as $\ell_p$ with $p>2$, and analyzes how these constants relate to classical invariants like $\mathcal{T}(\mathcal{X})$, $\mathcal{C'_{NJ}}(\mathcal{X})$, and $\delta_\mathcal{X}(\varepsilon)$. The work uses these constants to characterize uniformly non-square spaces and spaces with normal structure, and ties the two new constants to Hilbert-space geometry through exact values in Hilbert spaces. Collectively, the results provide new geometric tools for understanding Banach-space structure and the interplay between non-squareness, uniform convexity, and Hilbert-space characterization.

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(λ,μ,X ) and T2(\k{appa},τ,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.

Figures (1)

• Figure 1: The vector diagram used to prove that $\mathcal{X}$ is uniformly non-square.

Theorems & Definitions (48)

• Definition 1
• Remark 1
• Lemma 1
• proof
• Proposition 1
• proof
• Proposition 2
• proof
• Example 1
• Lemma 2