Abstract
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
→
A
→
G
→
T
→
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].