The fixed point property via dual space properties

A Banach space has the weak fixed point property if its dual space has a weak$^*$ sequentially compact unit ball and the dual space satisfies the weak$^*$ uniform Kadec-Klee property; and it has the \fpp if there exists $\epsilon>0$ such that, for every infinite subset $A$ of the unit sphere of the dual space, $A\cup (-A)$ fails to be $(2-\epsilon)$-separated. In particular, $E$-convex Banach spaces, a class of spaces that includes the uniformly nonsquare spaces, have the fixed point property.