User:Nsoum/Sternberg convexity theorem

In mathematics, the Atiyah / Guillemin - Sternberg theorem, proved independently by Michael Atiyah and Victor Guillemin - Shlomo Sternberg, states that for a compact and connected Hamiltonian $$\mathbb{T}^n$$-manifold the image of a moment map is a convex polytope. It also explicitly describes the vertices of the polytope. The AGS theorem is a generalization of the Schur-Horn theorem. This result was further extended by Frances Kirwan to the context of non-abelian compact Lie group actions which is known as Kirwan convexity theorem.

Statement of the theorem
Let $$\mathbb{T}^n$$ be the $$n$$ - dimensional torus group and let $$M$$ be a compact and connected manifold with a Hamiltonian $$\mathbb{T}^n$$ - action with moment map $$\Phi : M \rightarrow \mathfrak{t}^*$$, where $$\mathfrak{t}$$ is the Lie algebra of $$\mathbb{T}^n$$. Then $$\Phi(M)$$ is the convex polytope generated by the set $$\{ \Phi(p) : p \in M, t \cdot p = p \; \;\forall \;t \in \mathbb{T}^n\}$$. In other words, $$\Phi(M)$$ is the convex polytope generated by the image under $$\Phi$$ of the fixed - point set of the $$\mathbb{T}^n$$ - action on $$M$$.