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Dedicated to the memory of Moez Alimohamed

Given a Hamiltonian torus action the image of the momentum map is a convex polytope; this famous convexity theorem of Atiyah, Guillemin and Sternberg still holds when the action is not required to be Hamiltonian. Our generalization of the convexity theorem states that, given a symplectic torus action, the momentum map can be defined on an appropriate covering of the manifold and has as image a convex polytope along a rational subspace times its orthogonal space. We also prove stability of this property under small perturbations of the symplectic structure. The technique developed allows us to extend the result to any compact group action and also to deduce that any symplectic torus action, with fixed points, on a manifold of double the dimension of the torus, is Hamiltonian.