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 $G$-manifold is a stratified symplectic space. Suppose $1\to A\to G\to T\to 1$ is an exact sequence of compact Lie groups and $T$ is a torus. Then the reduction of a Hamiltonian $G$-manifold with respect to $A$ yields a Hamiltonian $T$-space. We show that if the $A$-moment map is proper, then the convexity theorem holds for such a Hamiltonian $T$-space, even when it is singular. We also prove that if, furthermore, the $T$-space has dimension $2\dim T$ and $T$ 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].