Convexity properties of gradient maps

We consider the action of a real reductive group G on a Kaehler manifold Z which is the restriction of a holomorphic action of the complexified group G^C. We assume that the induced action of a compatible maximal compact subgroup U of G^C on Z is Hamiltonian. We have an associated gradient map obtained from a Cartan decomposition of G. For a G-stable subset Y of Z we consider convexity properties of the intersection of the image of Y under the gradient map with a closed Weyl chamber. Our main result is a Convexity Theorem for real semi-algebraic subsets Y of the projective space corresponding to a unitary representation of U.