Convex viscosity solutions and state constraints

We establish the convexity of a viscosity solution of some general second order fully nonlinear elliptic equation with state constraints boundary conditions. Our method combines a comparison principle with the observation that, under suitable assumptions, the convex envelope of the solution is a supersolution. This property relies on the characterization of the viscosity subjet of the convex envelope of a lower semicontinuous coercive function. The equation solved by the conjugate of a convex solution as well as partial convexity are topics we also discuss.