Abstract: Atiyah's well known convexity theorem states that for a Hamiltonian action by the torus T on a compact connected symplectic manifold M the image \Phi(M) under the associated moment map \Phi is convex. Duistermaat in addition considered antisymplectic involutions \tau on M such that \Phi is invariant under \tau. He showed that \Phi(M)=\Phi(Q) for Lagrangian submanifolds Q that arise as fixed point sets of such involutions. We prove a generalization of Duistermaat's symplectic convexity theorem for involutions \tau which satisfy several compatibility properties with the torus action, but which are not necessarily antisymplectic. By the same method we can also extend a Duistermaat-type theorem for non-compact M with proper \Phi. All the symplectic convexity theorems mentioned have applications to the structure theory of semisimple Lie groups. With the generalization of Duistermaat's theorem we are able to extend existing symplectic proofs for Kostant's and Neeb's convexity theorems to all semisimple groups. In addition we develop the symplectic framework to refine a recently discovered complex convexity theorem.