The Dunkl-Williams constant related to Birkhoff orthogonality in Banach spaces (Q6551375)

\textit{A. Jiménez-Melado} et al. [J. Math. Anal. Appl. 342, No. 1, 298--310 (2008; Zbl 1151.46009)] proposed the Dunkl-Williams constant as follows:\N\[\NDW(X) = \sup \left\{ \frac{\|x\| + \|y\|}{\|x - y\|} \bigg\|\frac{x}{\|x\|}-\frac{y}{\|y\|}\bigg\| : x, y \in X \setminus \{0\}, x \neq y \right\},\N\]\N\NPreviously in 1993, \textit{A. M. Al-Rashed} [J. Math. Anal. Appl. 176, No. 2, 587--593 (1993; Zbl 0895.46011)] introduced the following:\N\[\N\psi_p(X) = \sup \left\{ \frac{\left( \|x\|^p + \|y\|^p \right)^{\frac{1}{p}}}{\|x - y\|}\bigg\|\frac{x}{\|x\|}-\frac{y}{\|y\|}\bigg\| : x, y \in X \setminus \{0\}, x \neq y \right\}, \small \quad \text{where } p > 0.\N\]\N\NClearly, \(\Psi_p(X)\) can be regarded as a generalization of \(DW(X)\). In this paper the authors have introduced a different version of the above constants, specifically related to Birkhoff-James orthogonality. These definitions are given as follows:\N\[\NDW_B(X) = \sup \left\{ \frac{\|x\| + \|y\|}{\|x - y\|} \bigg\|\frac{x}{\|x\|}-\frac{y}{\|y\|}\bigg\| : x, y \in X \setminus \{0\}, \, x \perp_B y \right\}\N\]\Nand\N\[\NDW_B^p(X) = \sup \left\{ \frac{\left(\|x\|^p + \|y\|^p\right)^{1/p}}{\|x - y\|}\bigg\|\frac{x}{\|x\|}-\frac{y}{\|y\|}\bigg\| : x, y \in X \setminus \{0\}, \, x \perp_B y \right\},\N\]\Nwhere \(p > 0\). The authors have demonstrated that the upper bounds of \(DW_{B}(X)\) and \(DW(X)\) are different. In this paper they have explored connections between these constants and other known constants. They have obtained characterizations of Hilbert spaces and uniformly non-square Banach spaces. Additionally, they have examined the Radon planes via \(DW_B(X)\) and obtained a characterization of affine regular hexagonal unit sphere. In particular, the value of \(DW_{B}^p(\ell_{\infty}-\ell_1)\) has been estimated.