Degenerations of convex projective structures via Hausdorff limits

A properly convex projective orbifold is the quotient of a bounded convex domain, endowed with its Hilbert geometry, by a discrete group of isometries. In the case that the convex projective orbifold is closed (compact without boundary), a classical theorem of Benoist states that the domain cannot contain a codimension-1 face apart from the simplex case. In the context of a degeneration of strictly convex projective structures on a closed orbifold, i.e. a sequence $\Omega_n/\Gamma_n$ not converging to a convex projective orbifold, we study the Hausdorff limit $\Omega_\infty$ of $\Omega_n$ as a sequence of domains. We produce examples of $\Omega_\infty$ not being strictly convex and, even more, being a polytope but not a simplex. Finally, we provide necessary and sufficient conditions for $\Omega_\infty$ not to contain a codimension-1 face.