  For a Banach space $B$ and for a class $\A$ of its bounded closed retracts,
endowed with the Hausdorff metric, we prove that retractions on elements $A \in
\A$ can be chosen to depend continuously on $A$, whenever nonconvexity of each
$A \in \A$ is less than $\f{1}{2}$. The key geometric argument is that the set
of all uniform retractions onto an $\a-$paraconvex set (in the spirit of E.
Michael) is $\frac{\a}{1-\a}-$paraconvex subset in the space of continuous
mappings of $B$ into itself. For a Hilbert space $H$ the estimate
$\frac{\a}{1-\a}$ can be improved to $\frac{\a (1+\a^{2})}{1-\a^{2}}$ and the
constant $\f{1}{2}$ can be reduced to the root of the equation $\a+
\a^{2}+a^{3}=1$.
