  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 $1\ra A\ra G\ra T\ra 1$
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 $2dim
T$ 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.
