A renorming of `1, explored here in detail, shows that the copies of `1 produced in the proof of the Kadec-Pe lczynski theorem inside nonreflexive subspaces of L1[0, 1] cannot be produced inside general nonreflexive spaces that contain copies of `1. Put differently, James’s distortion theorem producing oneplus-epsilon-isomorphic copies of `1 inside any isomorphic copy of `1 is, in a certain sense, optimal. A similar renorming of c0 shows that James’s distortion theorem for c0 is likewise optimal. James’s distortion theorems for `1, the space of absolutely summable sequences of scalars, and c0, the space of null sequences of scalars, are well-known [J]. The former states that, whenever a Banach space contains a subspace isomorphic to `1, the Banach space contains subspaces that are almost isometric to `1. Several of the authors of this article, individually and in concert, have tried to use this feature of `1 to determine if all (equivalent) renormings of `1 fail to have the fixed point property for nonexpansive mappings (the FPP); i.e. if, in any renorming of `1, there exist a nonempty, closed, bounded and convex subset C and a nonexpansive self-map T of C without a fixed point. The basis of these attempts was to use the fact that `1 in its usual norm fails to have the fixed point property and, since each renorming of `1 contains subspaces almost isometric to `1, a perturbation of the usual example would hopefully produce a nonexpansive self-map of a nonempty, closed, bounded, convex set in any renorming of `1. Similar attempts in c0 were also made. What appeared to be needed in these attempts were strengthened versions of James’s distortion theorems. To be specific, James’s theorem for `1 states that if a Banach space X with norm ‖ · ‖ contains an isomorphic copy of `1, then, for each > 0, there exists a sequence (xk) in the unit sphere of X such that (1− ) ∑∞ k=1 |tk| ≤ ‖ ∑∞ k=1 tkxk‖ ≤ ∑∞ k=1 |tk| for all (tk) ∈ `1. The proof of the theorem shows even more than the statement indicates. The sequence (xk) may be chosen to have the additional property that, if ( n) is a sequence of positive numbers decreasing to 0, then for each n, (1 − n) ∑∞ k=n |tk| ≤ ‖ ∑∞ k=n tkxk‖ ≤ ∑∞ k=n |tk| , for all (tk) ∈ `1. That is, for each δ > 0, by ignoring a finite number of terms at the beginning of the sequence (xk), one obtains copies of `1 which are (1 + δ)-isomorphic to `1. This Received by the editors May 8, 1995 and, in revised form, July 7, 1995. 1991 Mathematics Subject Classification. Primary 46B03, 46B20.