  For the Lie algebra $\g$ of a connected infinite-dimensional Lie group~$G$,
there is a natural duality between so-called semi-equicontinuous weak-*-closed
convex Ad^*(G)-invariant subsets of the dual space $\g'$ and Ad(G)-invariant
lower semicontinuous positively homogeneous convex functions on open convex
cones in $\g$. In this survey, we discuss various aspects of this duality and
some of its applications to a more systematic understanding of open invariant
cones and convexity properties of coadjoint orbits. In particular, we show that
root decompositions with respect to elliptic Cartan subalgebras provide
powerful tools for important classes of infinite Lie algebras, such as
completions of locally finite Lie algebras, Kac--Moody algebras and twisted
loop algebras with infinite-dimensional range spaces. We also formulate various
open problems.
