Given a local diffeomorphism ƒ : ℝn → ℝn, we consider certain in- compressibility conditions on the parallelepiped Dƒ(x) ([0, 1]n) which imply that the pre-image of an affine subspace is non-empty and has trivial homotopy groups. These conditions are then used to establish criteria for ƒ to be globally invertible, generalizing in all dimensions the previous results of M. Sabatini.