  We prove, in a geometric way, that the standard contact structure on the real
projective space of dimension $2n-1$ is not Liouville fillable for $n \ge 3$
and odd. We also prove that, for all $n$, semipositive fillings of those
contact structures are simply connected. Finally we give yet another proof of
the Eliashberg-Floer-McDuff theorem on the diffeomorphism type of the
symplectically aspherical fillings of the standard contact structure on the
$(2n-1)$-dimensional sphere.
