Remarks on the abelian convexity theorem

TL;DR

This paper discusses generalizations of abelian convexity theorems, providing simplified proofs and applications to actions on probability measures using Kempf-Ness functions and convexity concepts.

Contribution

It offers new insights and streamlined proofs for classical convexity theorems and extends the application to actions on probability measures.

Findings

Convexity along orbits established in a general setting.

Short proofs of Atiyah-Guillemin-Sternberg theorem provided.

Application to actions on probability measures demonstrated.

Abstract

This note contains some observations on abelian convexity theorems. 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.