On the calculation of the Dunkl-Williams constant of normed linear spaces
Let \(X\) be a normed space whose dimension is greater than one. \textit{C. F. Dunkl} and \textit{K. S. Williams} [Am. Math. Mon. 71, 53--54 (1964; Zbl 0178.16202)] proved that \(\left\|\frac{x}{\|x\|}-\frac{y}{\|y\|}\right\|\leq 4\frac{\|x-y\|}{\|x\|+\|y\|}\) for all nonzero elements \(x,y\in X\). The constant \(4\) in the inequality can be replaced by \(2\) if \(X\) is an inner product space. (In fact, the latter gives a characterization of inner product spaces.) Thus, \textit{A. Jiménez-Melado, E. Llorens-Fuster} and \textit{E. M. Mazcuñán-Navarro} [J. Math. Anal. Appl. 342, No. 1, 298--310 (2008; Zbl 1151.46009)] introduced the `Dunkl-Williams constant' as follows: \(DW(X):=\sup\{\frac{\|x\|+\|y\|}{\|x-y\|}\left\|\frac{x}{\|x\|}-\frac{y}{\|y\|}\right\|:x,y\in X\setminus\{0\},\,x\neq y\}\). In the present paper, the authors give some characterizations of this constant and calculate \(DW(\ell_2\text{-}\ell_\infty)\). Recall that \(\ell_2\text{-}\ell_\infty\) is the two-dimensional space \(\mathbb{R}^2\) equipped with the \(\ell_2\) norm for \(x\) in the first and the third quadrants, and with the \(\ell_\infty\) norm for \(x\) in the second and fourth quadrants.
scientific article; zbMATH DE number 6176175
scientific article