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User:Nsoum/Sternberg convexity theorem

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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 -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

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Let be the - dimensional torus group and let be a compact and connected manifold with a Hamiltonian - action with moment map , where is the Lie algebra of . Then is the convex polytope generated by the set . In other words, is the convex polytope generated by the image under of the fixed - point set of the - action on .

References

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