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.