TITLE: Atiyah-Guillemin-Sternberg convexity theorem
QUESTION [5 upvotes]: I would like to study the Atiyah-Guillemin-Sternberg convexity theorem: proof and applications. I am already familiarised with hamiltonian actions, moment maps...and with elementary Morse theory.
So my problem is to find a detailed proof of this theorem:

What are the prerequisits: Morse-Bott functions, equivariant Darboux
Theorem...?
Is the original proof by Atiyah different from the Guillemin-Sternberg's proof?
What is "the best reference" for a detailed treatment of this theorem?

Thanks for any help.

REPLY [4 votes]: 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.