Thanks Thomas, Liviu and Olga, The Atiyah's proof is done by induction on the dimension of the torus. If $\mu$ is the moment map and

$A_m$: the level sets of $\mu$ are connected, for any $\mathbb{T}^m$ hamiltonian action.

 

$B_m$: the image of $\mu$ is convex, for any $\mathbb{T}^m$ hamiltonian action.

The hard part (for me) is $A_1$ which is based on the connectedness of the levels of a Morse-Bott function on a compact manifold!

The rest of the proof is very well explained in:

1. Ana Cannas da Silva, Lectures on Symplectic Geometry (as exercises),
2. Michèle Audin: Topology of torus actions on symplectic manifolds,
3. http://www.math.nyu.edu/~kessler/teaching/group/convexity.pdf

The book by Liviu Nicolaescu is very useful and the complete proof can be found in:

McDuff & Salamon, Introduction to Symplectic Topology.

Thanks Thomas, Liviu and Olga, The Atiyah's proof is done by induction on the dimension of the torus. If $\mu$ is the moment map and

$A_m$: the level sets of $\mu$ are connected, for any $\mathbb{T}^m$ hamiltonian action.

$B_m$: the image of $\mu$ is convex, for any $\mathbb{T}^m$ hamiltonian action.

The hard part (for me) is $A_1$ which is based on the connectedness of the levels of a Morse-Bott function on a compact manifold!

The rest of the proof is very well explained in:

1. Ana Cannas da Silva, Lectures on Symplectic Geometry (as exercises),
2. Michèle Audin: Topology of torus actions on symplectic manifolds,
3. http://www.math.nyu.edu/~kessler/teaching/group/convexity.pdf

The book by Liviu Nicolaescu is very useful and the complete proof can be found in:

McDuff & Salamon, Introduction to Symplectic Topology.