  The main result of this paper is that every non-reflexive subspace $Y$ of
$L_1[0,1]$ fails the fixed point property for closed, bounded, convex subsets
$C$ of $Y$ and nonexpansive (or contractive) mappings on $C$. Combined with a
theorem of Maurey we get that for subspaces $Y$ of $L_1[0,1]$, $Y$ is reflexive
if and only if $Y$ has the fixed point property. For general Banach spaces the
question as to whether reflexivity implies the fixed point property and the
converse question are both still open.
