The skew James type constant in Banach spaces
Zhiyong Rao, Qi Liu, Qiong Wu, Zhouping Yin, Qichuan Ni
TL;DR
The paper introduces a skew James type constant $\mathcal{J}_t[\tau,\mathcal{X}]$ in Banach spaces and studies its representations, convexity, and interaction with the modulus of convexity via $\delta_{\mathcal{X}}(\varepsilon)$. It also defines $\mathcal{G}_t(\mathcal{X})$ and derives bounds connecting these constants to classical ones like the James constant $\mathcal{J}(\mathcal{X})$ and the Zbaganu constant $\mathcal{C}_{\mathcal{Z}}(\mathcal{X})$. Main results include equivalent representations for $t\ge1$, two-dimensional subspace reductions, and monotonic relationships with the parameter $t$, plus a bound for $\mathcal{G}_{-\infty}(\mathcal{X})$ and an explicit example showing $\mathcal{G}_{-\infty}(\mathcal{X})<\mathcal{C}_{\mathcal{Z}}(\mathcal{X})$ in the Day–James space. The work contributes a skewed geometric framework for Banach-space analysis with potential implications for uniform non-squareness and fixed-point properties.
Abstract
In the past, Takahashi has introduced the James type constants $\mathcal{J}_{\mathcal{X} ,t}(τ)$. Building upon this foundation, we introduce an innovative skew James type constant, denoted as $\mathcal{J}_t[τ,\mathcal{X}]$, which is perceived as a skewed counterpart to the traditional James type constants. We delineate a novel constant, and proceed to ascertain its equivalent representations along with certain attributes within the context of Banach spaces, and then an investigation into the interrelation between the skewness parameter and the modulus of convexity is conducted, after that we define another new constant $\mathcal{G}_t(\mathcal{X})$, and some conclusions were drawn.