  We prove an analogue of Kostant's convexity theorem for split real and
complex Kac-Moody groups associated to free and cofree root data. The result
can be seen as a first step towards describing the multiplication map in a
Kac-Moody group in terms of Iwasawa coordinates. Our method involves a detailed
analysis of the geometry of Weyl group orbits in the Cartan subalgebra of a
real Kac-Moody algebra. It provides an alternative proof of Kostant convexity
for semisimple Lie groups and also generalizes a linear analogue of Kostant's
theorem for Kac-Moody algebras that has been established by Kac and Peterson in
1984.
