Reduced Delzant spaces and a convexity theorem

The convexity theorem of Atiyah and Guillemin–Sternberg says that any connected compact manifold with Hamiltonian torus action has a moment map whose image is the convex hull of the image of the fixed point set. Sjamaar–Lerman proved that the Marsden–Weinstein reduction of a connected Hamitonian GG-manifold is a stratified symplectic space. Suppose 1→A→G→T→11→A→G→T→1 is an exact sequence of compact Lie groups and TT is a torus. Then the reduction of a Hamiltonian GG-manifold with respect to AA yields a Hamiltonian TT-space. We show that if the AA-moment map is proper, then the convexity theorem holds for such a Hamiltonian TT-space, even when it is singular. We also prove that if, furthermore, the TT-space has dimension 2dimT and TT acts effectively, then the moment polytope is sufficient to essentially distinguish their homeomorphism type, though not their diffeomorphism types. This generalizes a theorem of Delzant in the smooth case. This paper is a concise version of a companion paper [B. Lian. B. Song, A convexity theorem and reduced Delzant spaces, math.DG/0509429].