Simple non-rational convex polytopes via symplectic geometry

In this article we consider a generalization of manifolds and orbifolds which we call quasifolds; quasifolds of dimension k are locally isomorphic to the quotient of the space R^k by the action of a discrete group — typically they are not Hausdorff topological spaces. The analogue of a torus in this geometry is a quasitorus. We define Hamiltonian actions of quasitori on symplectic quasifolds and we show that any simple convex polytope, rational or not, is the image of the moment mapping for a family of effective Hamiltonian actions on symplectic quasifolds having twice the dimension of the corresponding quasitorus.