  In our note we show the very close connection between the existence of a
Finite Dimensional Decomposition (FDD for short) for a separable Banach space
$X$ and the existence of a Lipschitz retraction of $X$ onto a small (in a
certain precise sense) generating convex and compact subset $K$ of $X$. In one
direction, if $X$ admits an FDD then we construct a Lipschitz retraction onto a
small generating convex and compact set $K$. On the other hand, we prove that
if $X$ admits a small generating compact Lipschitz retract then $X$ has the
$\pi$-property. We note that it is still unknown if the $\pi$-property is
isomorphically equivalent to the existence of an FDD. For dual Banach spaces
this is true, so our results lead in particular to a characterization of the
FDD property for dual Banach spaces $X$ in terms of the existence of Lipschitz
retractions onto small generating convex and compact subsets of $X$. It is
conceivable that our results will find applications in the area of Lipschitz
isomorphisms of Banach spaces. Our arguments make critical use of the
Lipschitzization of coarse Lipschitz mappings due to J. Bourgain, and of an
unpublished complementability result of V. Milman. We give an example of a
small generating convex compact set which is not a Lipschitz retract of
$C[0,1]$, although it is contained in a small convex Lipschitz retract and
contains another one. In the last part of our note we characterize
isomorphically Hilbertian spaces as those Banach spaces $X$ for which every
convex and compact subset is a Lipschitz retract of $X$. Finally, we prove that
a convex and compact set $K$ in any Banach space with a Uniformly Rotund in
Every Direction norm is a uniform retract, of every bounded set containing it,
via the nearest point map.
