Facultad de Matemáticas, Universidad de Sevilla, P.O. Box 1160, Sevilla 41080, Spain
Copyright © 2010 T. Domínguez Benavides. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.