Remarks on the Abelian convexity theorem

This paper contains some observations on abelian convexity the- orems. Convexity along an orbit is established in a very general setting using Kempf–Ness functions. This is applied to give short proofs of the Atiyah– Guillemin–Sternberg theorem and of abelian convexity for the gradient map in the case of a real analytic submanifold of complex projective space. Finally we give an application to the action on the probability measures.