  Let K be a connected Lie group and M a Hamiltonian K-manifold. In this paper,
we introduce the notion of convexity of M. It implies that the momentum image
is convex, the moment map has connected fibers, and the total moment map is
open onto its image. Conversely, the three properties above imply convexity. We
show that most Hamiltonian manifolds occuring "in nature" are convex (e.g., if
M is compact, complex algebraic, or a cotangent bundle). Moreover, every
Hamiltonian manifold is locally convex. This is an expanded version of section
2 of my paper dg-ga/9712010 on Weyl groups of Hamiltonian manifolds.
