Convex co-compact actions of relatively hyperbolic groups

In this paper we consider discrete groups in ${\rm PGL}_d(\mathbb{R})$ acting convex co-compactly 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-Gu\'eritaud-Kassel and is analogous to a result of Hruska-Kleiner for ${\rm CAT}(0)$ spaces.