A weighted Birkhoff orthogonal James-type constant

Abstract

Let $X$ be a real Banach space and $\hat{I}»\in [0,1]$. Motivated by orthogonal versions of the James constant, we introduce the weighted Birkhoff orthogonal James-type constant $$J_{\hat{I}}{»}^{\perp }(X)=\sup \left\{\min \{\Vert \hat{I}»x+(1-\hat{I}»)y\Vert ,\Vert \hat{I}»x-(1-\hat{I}»)y\Vert \}:x,y\in S_X,x{\perp }_{B}y\right\},$$ where \(λ\in[0,1]\) 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 $J_{\hat{I}}{»}^{\perp }(X)$ with respect to $\hat{I}»$ is proved. New characterizations of uniformly nonsquare spaces are obtained; in particular, $J_{\hat{I}}{»}^{\perp }(X)=1$ for some $\hat{I}»\in (0,1)$ 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.