A new moduli for Banach spaces, Dunkl-Williams constant and Holder continuity of radial projection

Let $X$ be a Banach space and let $DW(X)$ denotes the Dunkl-Williams constant of $X$. In this paper, we first introduce moduli $ \gamma_{X}:[0,2] \to [0,1]$ and show that $DW(X)=\sup\limits_{0<\epsilon \leq 2}\frac{\epsilon}{\gamma_{X}(\epsilon)}$, which provides a simple formula for calculating $DW(X)$ in terms of $\gamma_{X}(\epsilon)$. Then, we compute $\gamma_{H}(\epsilon)$, where $H$ is a Hilbert space, $\gamma_{l_{\infty}-l_1}(\epsilon)$, $\gamma_{l_{2}^{\infty}}(\epsilon)$ and we conclude that $DW(l_{\infty}-l_1)=1+\sqrt{2}$. Furthermore, we obtain a generalized Dunkl-Williams inequality and we conclude that, for each $\alpha\in (0,1)$, the radial projection on $X$ is $\alpha$-H\"older continuous. Mathematics Subject Classification (2010). 46B20.