A convexity theorem for Poisson actions of compact Lie groups

When a Poisson Lie group acts on a symplectic manifold there is a momentum mapping to the dual group. We prove that the image of this momentum mapping meets the positive Weyl chamber in the exponential of a convex set. The result adapts the well-known theorem of Guillemin, Sternberg, and Kirwan to a ''multiplicative'' (rather than ''additive'') setting.