  In this note, we state and give the main ideas of the proof of a real
convexity theorem for group-valued momentum maps. This result is a
quasi-Hamiltonian analogue of the O'Shea-Sjamaar theorem in the usual
Hamiltonian setting. We prove here that the image under the momentum map of the
fixed-point set of a form-reversing involution defined on a quasi-Hamiltonian
space is a convex polytope, that we describe as a subpolytope of the momentum
polytope.
