Distortion and stability of the fixed point property for non-expansive mappings

Let XX be a Banach space. We say that XX satisfies the fixed point property (weak fixed point property) if every non-expansive mapping defined from a convex closed bounded (convex weakly compact) subset of XX into itself has a fixed point. We say that XX satisfies the stable fixed point property (stable weak fixed point property) if the same is true for every equivalent norm which is close enough to the original one. Denote by P(X)P(X) the set formed by all equivalent norms with the topology of the uniform convergence on the unit ball of XX. We prove that the subset of P(X)P(X) formed by the norms failing the fixed point property is dense in P(X)P(X) when XX is a non-distortable space which fails the fixed point property. In particular, no renorming of ℓ1ℓ1 can satisfy the stable fixed point property. Furthermore, we show some examples of distortable spaces failing the weak fixed point property, which can be renormed to satisfy the stable weak fixed point property. As a consequence we prove that every separable Banach space can be renormed to satisfy the stable weak fixed point property.