New families of nonreflexive Banach spaces with the fixed point property

In this paper we define the concept of sequentially separating norm and we relate it to the fulfillment of the fixed point property for nonexpansive mappings. We develop a technique to construct families of nonreflexive Banach spaces with the fixed point property. Different examples will be exhibited. We also study the geometry of the Banach spaces that can be renormed with a sequentially separating norm.