  In this paper we show that many projective Anosov representations act convex
cocompactly on some properly convex domain in real projective space. In
particular, if a non-elementary word hyperbolic group is not commensurable to a
non-trivial free product or the fundamental group of a closed hyperbolic
surface, then any projective Anosov representation of that group acts convex
cocompactly on some properly convex domain in real projective space. We also
show that if a projective Anosov representation preserves a properly convex
domain, then it acts convex cocompactly on some (possibly different) properly
convex domain.
  We then give three applications. First, we show that Anosov representations
into general semisimple Lie groups can be defined in terms of the existence of
a convex cocompact action on a properly convex domain in some real projective
space (which depends on the semisimple Lie group and parabolic subgroup). Next,
we prove a rigidity result involving the Hilbert entropy of a projective Anosov
representation. Finally, we prove a rigidity result which shows that the image
of the boundary map associated to a projective Anosov representation is rarely
a $C^2$ submanifold of projective space. This final rigidity result also
applies to Hitchin representations.
