Symmetric form geometric constant related to isosceles orthogonality in Banach spaces
Qichuan Ni, Qi Liu, Yuxin Wang, Jinyu Xia, Ranran Wang
TL;DR
The article introduces the symmetric geometric constant $L_X(t)$ as an orthogonal perspective on the von Neumann-Jordan constant for Banach spaces, and establishes its fundamental properties, including sharp bounds, convexity, and continuity. It shows $L_X(t)=\tfrac{1}{2}\gamma_X(1-2t)$, linking the new constant to the classical $\,\gamma_X(t)$ and providing a direct route to $C_{NJ}(X)$ via $L_X$, enabling tight relations to $J(X)$, the modulus of convexity, and other geometric constants. The work computes explicit forms of $L_X(t)$ in common spaces such as $l_p$ and $l_p-l_q$, identifies uniform non-square spaces, and derives Hilbert-space characterizations through lower bounds of $L_X(t)$. It further connects $L_X(t)$ to uniform normal/strict structures and uniform smoothness, offering a comprehensive framework for understanding Banach space geometry from an orthogonal viewpoint and yielding exact results in several two-dimensional cases. Overall, the paper provides new tools for analyzing geometric properties, establishing precise links between orthogonality-based constants and classical space classifications.
Abstract
In this article, we introduce a novel geometric constant $L_X(t)$, which provides an equivalent definition of the von Neumann-Jordan constant from an orthogonal perspective. First, we present some fundamental properties of the constant $L_X(t)$ in Banach spaces, including its upper and lower bounds, as well as its convexity, non-increasing continuity. Next, we establish the identities of $L_X(t)$ and the function $γ_X(t)$, the von Neumann-Jordan constant, respectively. We also delve into the relationship between this novel constant and several renowned geometric constants (such as the James constant and the modulus of convexity). Furthermore, by utilizing the lower bound of this new constant, we characterize Hilbert spaces. Finally, based on these findings, we further investigate the connection between this novel constant and the geometric properties of Banach spaces, including uniformly non-square, uniformly normal structure, uniformly smooth, etc.