  Let M be a complete simply connected Riemannian manifold, with sectional
curvature K bounded above by -1. Under some assumptions on the geometry of the
boundary of M, which are satisfied for instance if M is a symmetric space, or
has dimension 2, we prove that given any family of horoballs in M, and any
point x_0 outside these horoballs, it is possible to shrink uniformly, by a
finite amount depending only on M, these horoballs so that some geodesic ray
starting from x_0 avoids the shrunk horoballs. As an application, we give a
uniform upper bound on the infimum of the heights of the closed geodesics in
the finite volume quotients of M.
