Symmetric form geometric constant related to isosceles orthogonality in Banach spaces

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 $\gamma_X(t)$, 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.