We analyze some convexity properties of the image maps on symplectic cones, similar to the ones obtained by Guillemin-Sternberg and Atiyah for compact symplectic manifolds in the early 80's. We prove the image of the moment map associated to the symplectic action of an n-torus on a symplectic cone is a polytopic convex cone in R^n. Then, we generalize these results to symplectic manifolds obtained by special perturbations of the symplectic structure of a cone: we obtain sufficient (and essentially necessary) conditions for the image of the moment map associated to the perturbed form to remain unchanged.