  A natural way of generalising Hamiltonian toric manifolds is to permit the
presence of generic isolated singularities for the moment map. For a class of
such ``almost-toric 4-manifolds'' which admits a Hamiltonian $S^1$-action we
show that one can associate a group of convex polygons that generalise the
celebrated moment polytopes of Atiyah, Guillemin-Sternberg. As an application,
we derive a Duistermaat-Heckman formula demonstrating a strong effect of the
possible monodromy of the underlying integrable system.
