In 2017, Danciger-Gue'ritaud-Kassel (DGK) introduced a `higher-rank' generalization of Kleinian groups using a notion of convex co-compactness for groups acting on Hilbert geometries (properly convex subsets of RP^n). DGK (and independently, Zimmer) proved that for hyperbolic groups, admitting such a convex co-compact action is equivalent to the ``convex-core'' being strictly convex. In their paper, DGK had asked whether an analogous result holds for relatively hyperbolic groups. We provide a complete answer to this question by introducing the notion of Hilbert geometries with isolated simplices. This is joint work with Andrew Zimmer. Speaker(s): Mitul Islam (U Michigan)