A global shadow lemma and logarithm law for geometrically finite Hilbert geometries

For geometrically finite group actions on hyperbolic metric spaces and under certain assumptions on the growth of parabolic subgroups, we prove a global shadow lemma for Patterson–Sullivan measures, as well as a Dirichlet-type theorem and a logarithm law for excursion of geodesics into cusps. We then apply these results to geometrically finite quotients of strictly convex Hilbert geometries with C1 boundary.