The fixed point property under renorming in some classes of Banach spaces

Assume that YY is a Banach space such that R(Y)<2R(Y)<2 where R(⋅)R(⋅) is García-Falset’s coefficient, and XX is a Banach space which can be continuously embedded in YY. We prove that XX can be renormed to satisfy the weak Fixed Point Property (w-FPP). On the other hand, assume that KK is a scattered compact topological space such that K(ω)=0̸K(ω)=0̸ and C(K)C(K) is the space of all real continuous functions defined on KK with the supremum norm. We will show that C(K)C(K) can be renormed to satisfy R(C(K))<2R(C(K))<2. Thus, both results together imply that any Banach space which can be continuously embedded in C(K)C(K), KK as above, can be renormed to satisfy the w-FPP. These results extend a previous one about the w-FPP under renorming for Banach spaces which can be continuously embedded in c0(Γ)c0(Γ). Furthermore, we consider a metric in the space PP of all norms in C(K)C(K) which are equivalent to the supremum norm and we show that for almost all norms in PP (in the sense of porosity) C(K)C(K) satisfies the w-FPP.