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:
- Ana Cannas da Silva, Lectures on Symplectic Geometry (as exercises),
- Michèle Audin: Topology of torus actions on symplectic manifolds,
- 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.
