Co-existence of the occupied and vacant phase in Boolean models in three or more dimensions

Consider a continuum percolation model in which, at each point of a d-dimensional Poisson process of rate λ, a ball of radius 1 is centred. We show that, for any d≧3, there exists a phase where both the regions, occupied and vacant, contain unbounded components. The proof uses the concept of enhancement for the Boolean model, and along the way we prove that the critical intensity of a Boolean model defined on a slab is strictly larger than the critical intensity of a Boolean model defined on the whole space.