Asymptotically Isometric Copies of c and Renormings 0 of Banach Spaces

If a Banach space X contains an asymptotically isometric copy of c0 , then X fails to have weak normal structure. Consequently, if X is an infinite-dimensional subspace of c0 ,5?5`., then X fails to have weak normal structure. Also, every equivalent renorming of c0 G., for G uncountable, fails to have weak normal structure. Every separable Banach space can be equivalently renormed so as not to contain an asymptotically isometric copy of c0. Every Banach space with the generalized Gossez-Lami Dozo GGLD.property fails to contain a subspace isomorphic to c0.