On nearly isometric l1 -preduals

ABSTRACT
The notion of nearly isometric Banach spaces appeared for the first time in the celebrated book by Stefan Banach Th´eorie des op´erations lin´eaires, Warsaw, 1932. Recall that two isomorphic Banach spaces X and Y are said to be nearly isometric (or almost isometric) if d(X, Y ) := inf { || Ø || || Ø^-1 ||: Ø   is an isomorphism from X onto Yo = 1.
For every predual X of `1 such that the standard basis in `1 is weak convergent, we give explicit models of all Banach spaces that are nearly isometric to X. We also present some illustrations and applications of our considerations.