On Banach space properties that are not invariant under the Banach-Mazur distance 1

We consider which properties of Banach spaces are not invariant under the Banach-Mazur distance 1; we will say that a given Property P is not invariant under the Banach-Mazur distance 1 if there exist two Banach spaces X and Y such that X has Property P, Y fails Property P, and the Banach-Mazur distance d(X,Y)=1. The main results of this paper concern the case of separable Lindenstrauss spaces. First, in the setting of preduals of ℓ1, we give some geometric equivalences for polyhedral properties. Next, we show that, even in the restricted framework of preduals of ℓ1, most of the notions of polyhedral Banach spaces (and their geometric equivalences) fail to be invariant under the Banach-Mazur distance 1. Then, in the general case of Banach spaces, we indicate some other geometrical properties that are invariant or not invariant under the Banach-Mazur distance 1.