  We extend the Local-to-Global-Principle used in the proof of convexity
theorems for momentum maps to not necessarily closed maps whose target space
carries a convexity structure which need not be based on a metric. Using a new
factorization of the momentum map, convexity of its image is proved without
local fiber connectedness, and for almost arbitrary spaces of definition.
Geodesics are obtained by straightening rather than shortening of arcs, which
allows a unified treatment and extension of previous convexity results.
