The Szlenk Index and the Fixed Point Property under Renorming

Assume that is a Banach space such that its Szlenk index is less than or equal to the first infinite ordinal . We prove that can be renormed in such a way that with the resultant norm satisfies , where is the García-Falset coefficient. This leads us to prove that if is a Banach space which can be continuously embedded in a Banach space with , then, can be renormed to satisfy the w-FPP. This result can be applied to Banach spaces which can be embedded in , where is a scattered compact topological space such that . Furthermore, for a Banach space , we consider a distance in the space of all norms in which are equivalent to (for which becomes a Baire space). If , we show that for almost all norms (in the sense of porosity) in , satisfies the w-FPP. For general reflexive spaces (independently of the Szlenk index), we prove another strong generic result in the sense of Baire category.