Some renormings with the stable fixed point property (Q2871099)

Let \(C\) be a subset of a Banach space \(X\) and \(T:C\to C\). We say that \(T\) is nonexpansive if \(\|Tx-Ty\|\leq\|x-y\|\) for all \(x,y\in C\). A Banach space \(X\) is said to have the weak fixed point property if every nonexpansive mapping defined on a weakly compact and convex subset of \(X\) has a fixed point. Let \(X\) be a Banach space and \(\lambda<\frac{\sqrt{33}-3}{2}\). The authors prove the following result: There is an equivalent norm \(|\cdot|\) on \(X\) such that \(Y\) has the weak fixed point property whenever the Banach-Mazur distance between a Banach space \(Y\) and \((X,|\cdot|)\) is less than \(\lambda\) if one of the following conditions holds: (1) \(X\) can be isomorphically embedded in a Banach space \(Z\) with an extended unconditional basis; (2) \(X\) is separable.