  We consider a closed convex set $C$ in a separable, infinite-dimensional
Hilbert space and endow the set $\mathcal{N}(C)$ of nonexpansive self-mappings
on $C$ with the topology of pointwise convergence. We introduce the notion of a
somewhat bounded set and establish a strong connection between this property
and the existence of fixed points for the generic $f\in\mathcal{N}(C)$, in the
sense of Baire categories. Namely, if $C$ is somewhat bounded, the generic
nonexpansive mapping on $C$ admits a fixed point, whereas if $C$ is not
somewhat bounded, the generic nonexpansive mapping on $C$ does not have any
fixed points. This results in a topological 0-1 law: the set of all
$f\in\mathcal{N}(C)$ with a fixed point is either meager or residual. We
further prove that, generically, there are no fixed points in the interior of
$C$ and, under additional geometric assumptions, we show the uniqueness of such
fixed points for the generic $f\in\mathcal{N}(C)$ and the convergence of the
iterates of $f$ to its fixed point.
