The difference $\frac12\dim(X)-\dim(T)$ is known as the complexity of the $T$-space (assumed effective), so that's the keyword you want to use. Such results as I've heard of are mainly for complexity one, by Yael Karshon and Sue Tolman:

• Centered complexity one Hamiltonian torus actions (2001);

• Complete invariants for Hamiltonian torus actions with two dimensional quotients (2003);

• Classification of Hamiltonian torus actions with two dimensional quotients (2011).

These papers may perhaps not quite have the focus you're asking for: rather than deduce information from the moment polytope alone, they are about enhancing it with extra data (the Duistermaat-Heckman measure, a genus and a "painting") so that the resulting invariant completely determines the $T$-space.

The third one also quotes some results on $S^1$-spaces of dimension 6, i.e. complexity 2.

The difference $\frac12\dim(X)-\dim(T)$ is known as the complexity of the $T$-space (assumed effective), so that's the keyword you want to use. Such results as I've heard of are mainly for complexity one, by Yael Karshon and Sue Tolman:

• Centered complexity one Hamiltonian torus actions (2001);

• Complete invariants for Hamiltonian torus actions with two dimensional quotients (2003);

• Classification of Hamiltonian torus actions with two dimensional quotients (2011).

These papers may perhaps not quite have the focus you're asking for: rather than deducing information from the moment polytope alone, they are about enhancing it with extra data (the Duistermaat-Heckman measure, a genus and a "painting") so that the resulting invariant completely determines the $T$-space.

Still, they should provide you with plenty of examples showing what information one cannot hope to obtain from the moment polytope alone.

The third one also quotes some results on $S^1$-spaces of dimension 6, i.e. complexity 2.

The difference $\frac12\dim(X)-\dim(T)$ is known as the complexity of the $T$-space (assumed effective), so that's the keyword you want to use. Such results as I've heard of are mainly for complexity one, by Yael Karshon and Sue Tolman:

• Centered complexity one Hamiltonian torus actions (2001);

• Complete invariants for Hamiltonian torus actions with two dimensional quotients (2003);

• Classification of Hamiltonian torus actions with two dimensional quotients (2011).

The third paper also quotes some results on $S^1$-spaces of dimension 6, i.e. complexity 2.

The difference $\frac12\dim(X)-\dim(T)$ is known as the complexity of the $T$-space (assumed effective), so that's the keyword you want to use. Such results as I've heard of are mainly for complexity one, by Yael Karshon and Sue Tolman:

• Centered complexity one Hamiltonian torus actions (2001);

• Complete invariants for Hamiltonian torus actions with two dimensional quotients (2003);

• Classification of Hamiltonian torus actions with two dimensional quotients (2011).

These papers may perhaps not quite have the focus you're asking for: rather than deduce information from the moment polytope alone, they are about enhancing it with extra data (the Duistermaat-Heckman measure, a genus and a "painting") so that the resulting invariant completely determines the $T$-space.

The third one also quotes some results on $S^1$-spaces of dimension 6, i.e. complexity 2.