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

We prove a global shadow lemma for Patterson-Sullivan measures in geometrically finite Hilbert manifolds. We also prove a Dirichlet-type theorem for hyperbolic metric spaces which have sufficiently regular Busemann functions. We apply these results to prove the logarithm law for excursion of geodesics into cusps in the setting of Hilbert geometries.