Lagrangian submanifolds and moment convexity

We consider a Hamiltonian torus action $T\times M \rightarrow M$ on a compact connected symplectic manifold $M$ and its associated momentum map $\Phi$. For certain Lagrangian submanifolds $Q\subseteq M$ we show that $\Phi (Q)$ is convex. The submanifolds $Q$ arise as the fixed point set of an involutive diffeomorphism $\tau :M\rightarrow M$ which satisfies several compatibility conditions with the torus action, but which is in general not anti-symplectic. As an application we complete a symplectic proof of Kostant's non-linear convexity theorem.