Convexity properties of the moment map for certain non-compact manifolds

Let M be a connected symplectic manifold on which a torus T acts in a Hamiltonian fashion and denote by f the corresponding moment map. If M is compact, the Atiyah, Guillemin- Sternberg theorem says that the set f(M) is convex; if M is noncompact this may not be the case. We show that if M is suitably convex, then f(M) can be explicitly described as the convex hull of a finite number of affine rays. As an application, we obtain a natural symplectic proof of the Paneitz convexity theorem for the positive Kaehler elliptic co-adjoint orbits.