  Consider a Hamiltonian action of a compact Lie group on a compact symplectic
manifold. A theorem of Kirwan's says that the image of the momentum mapping
intersects the positive Weyl chamber in a convex polytope. I present a new
proof of Kirwan's theorem, which gives explicit information on how the vertices
of the polytope come about and on how the shape of the polytope near any point
can be read off from infinitesimal data on the manifold. It also applies to
some interesting classes of noncompact or singular Hamiltonian spaces, such as
cotangent bundles and complex affine varieties.
