A weighted Birkhoff orthogonal James-type constant

Abstract

Let $X$ be a real Banach space and $\lambda \in[0,1]$. Motivated by orthogonal versions of the James constant, we introduce the weighted Birkhoff orthogonal James-type constant $$J_\lambda^{\perp}(X)=\sup \left\{\min \{\|\lambda x+(1-\lambda) y\|,\|\lambda x-(1-\lambda) y\|\}: x, y \in S_X, x \perp_B y\right\},$$ where \(\lambda\in[0,1]\) and $x \perp_B y$ stands for Birkhoff orthogonality. We establish its basic bounds, stability properties, and reduction principles, and clarify its relations with the orthogonal James constant $J_{\perp}(X)$. The 2 -Lipschitz continuity of $J_\lambda^{\perp}(X)$ with respect to $\lambda$ is proved. New characterizations of uniformly nonsquare spaces are obtained; in particular, $J_\lambda^{\perp}(X)=1$ for some $\lambda \in(0,1)$ if and only if $X$ is not uniformly nonsquare. We also discuss connections with strict convexity, uniform convexity, modulus of smoothness, and the von Neumann-Jordan constant.