Convex cocompact actions of relatively hyperbolic groups

We consider discrete groups in PGLd (R) acting convex cocompactly on a properly convex domain in real projective space. For such groups, we establish necessary and sufficient conditions for the group to be relatively hyperbolic in terms of the geometry of the convex domain. This answers a question of Danciger, Gueritaud and Kassel and is analogous to a result of Hruska and Kleiner for CAT(0) spaces.