  We investigate special lcs and twisted Hamiltonian torus actions on strict
lcs manifolds and characterize them geometrically in terms of the minimal
presentation. We prove a convexity theorem for the corresponding twisted moment
map, establishing thus an analog of the symplectic convexity theorem of Atiyah
and Guillemin-Sternberg. We also prove similar results for the symplectic
moment map (defined on the minimal presentation) whose image is then a convex
cone. In the special case of a compact toric Vaisman manifold, we obtain a
structure theorem.
