Let
X
X
be a separable Banach space. We give an almost characterization of the existence of a Finite Dimensional Decomposition (FDD for short) for
X
X
in terms of Lipschitz retractions onto generating compact subsets
K
K
of
X
X
.
In one direction, if
X
X
admits an FDD then we construct a Lipschitz retraction onto a small generating convex and compact set
K
K
. On the other hand, we prove that if
X
X
admits a “small” generating compact Lipschitz retract then
X
X
has the
π
\pi
-property. It is still unknown if the
π
\pi
-property is isomorphically equivalent to the existence of an FDD.
For dual Banach spaces this is true, so our results give a characterization of the FDD property for dual Banach spaces
X
X
.
We give an example of a small generating convex compact set which is not a Lipschitz retract of
C
[
0
,
1
]
C[0,1]
, although it is contained in a small convex Lipschitz retract and contains another one.
We characterize isomorphically Hilbertian spaces as those Banach spaces
X
X
for which every convex and compact subset is a Lipschitz retract of
X
X
.
Finally, we prove that a convex and compact set
K
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.