The skew generalized von Neumann-Jordan type constant in Banach spaces
Yuxin Wang, Qi Liu, Yueyue Feng, Jinyu Xia, Muhammad Sarfraz
TL;DR
This work introduces the skew generalized von Neumann-Jordan type constant $C_{-\infty}^p(\lambda,\mu,X)$ for real Banach spaces and develops a comprehensive set of properties, representations, and bounds, situating it among the classical constants $C_{NJ}^p(\lambda,\mu,X)$ and $J(X)$. It establishes sharp connections to isomorphic invariants via the Banach–Mazur distance and proves stability under renormings, highlighting the robustness of the constant under isomorphisms. A key contribution is linking $C_{-\infty}^p(\lambda,\mu,X)$ to the weak orthogonality coefficient $ω(X)$ to obtain a sufficient condition for normal structure, with implications for fixed-point theory of nonexpansive mappings. Collectively, the results enrich the geometric analysis of Banach spaces by providing new tools to characterize space geometry and structure through $C_{-\infty}^p(\lambda,\mu,X)$ and its relations to $J(X)$, $C_{NJ}^p(\lambda,\mu,X)$, and $ω(X)$.
Abstract
Recently, the von Neumann-Jordan type constants C(X) has defined by Takahashi. A new skew generalized constant Cp(λ,μ,X) based on C(X) constant is given in this paper. First, we will obtain some basic properties of this new constant. Moreover, some relations between this new constant and other constants are investigated. Specially, with the Banach-Mazur distance, we use this new constant to study isomorphic Banach spaces. Ultimately, by leveraging the connection between the newly introduced constant and the weak orthogonality coefficient ω(X), a sufficient condition for normal structure is established.