  The purpose of this paper is finding the essential attributes underlying the
convexity theorems for momentum maps. It is shown that they are of topological
nature; more specifically, we show that convexity follows if the map is open
onto its image and has the so called local convexity data property. These
conditions are satisfied in all the classical convexity theorems and hence they
can, in principle, be obtained as corollaries of a more general theorem that
has only these two hypotheses. We also prove a generalization of the
"Lokal-global-Prinzip" that only requires the map to be closed and to have a
normal topological space as domain, instead of using a properness condition.
This allows us to generalize the Flaschka-Ratiu convexity theorem to
non-compact manifolds.
