Strict convexity and $$C^1$$ regularity of solutions to generated Jacobian equations in dimension two

We present a proof of strict g-convexity in 2D for solutions generated Jacobian equations with g-Monge–Ampère measure bounded away from 0. Subsequently this implies $$C^1$$ differentiability the case above. Our follows one given by Trudinger and Wang Monge–Ampère case. Thus, like theirs, our argument is local yields quantitative estimate on g-convexity. As result new even optimal transport case: we weaken previously required domain convexity conditions. Moreover key assumptions, namely A3w convexity, are necessary.