  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.
