Renormings failing to have asymptotically isometric copies of ℓ1ℓ1 or c0c0

It is proved that every equivalent norm in ℓ1ℓ1 is the starting point of a ray of norms which fail to have an asymptotically isometric copy of ℓ1ℓ1. In the case where XX is separable, it is proved that the set of all equivalent norms in XX which fail to have an asymptotically isometric copy of ℓ1ℓ1 is residual. Similar results are obtained for the case of c0c0. In particular, we deduce that every separable Banach space has an equivalent norm for which it fails to contain an asymptotically isometric copy of ℓ1ℓ1 and c0c0.