Property($K^*$) Implies $R(X) \leq 1 + \frac{\displaystyle 1}{\displaystyle 1 + r_{X^*}(1)}$
Tim Dalby · arXiv (Cornell University) · 2021
It is shown that if the dual of a Banach space satisfies Property($K^*$) then $R(X) \leq 1 + \frac{\displaystyle 1}{\displaystyle 1 + r_{X^*}(1)} < 2$ where $r_{X^*}(c)$ is Opial's modulus for $X^*.$ Thus $X$ has the weak fixed point property.