A generalization of a renorming theorem by Lin and a new nonreflexive space with the fixed point property which is nonisomorphic to

Given 1 < p ≤ ∞ , we construct a Banach space X p hereditarily asymptotically isometric to l 1 , so that for n i , m k ∈ N it contains an isometric copy of ( ∑ i = 1 ∞ ∑ k = 1 m k l p n i ) 1 and X p ∗ contains a subspace isomorphic to c 0 ( Γ ) , where Γ has cardinality c , and use a generalization of a renorming theorem of Lin to renorm X p in order to have the fixed point property. Also, we give new renormings with the FPP in some known spaces. In particular we prove that ( c 0 , ‖ ⋅ ‖ α ) ∗ has the fixed point property, where ‖ ( x i ) ‖ α = sup i | x i | + α ∑ i | x i | 2 i , α > 0 . This gives an example of a space without the FPP whose dual (an isomorph of l 1 ) has the FPP.