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We present a proof of strict $g$-convexity in 2D for solutions of generatedJacobian equations with a $g$-Monge-Amp\`ere measure bounded away from 0.Subsequently this implies $C^1$ differentiability in the case of a$g$-Monge-Amp\`ere measure bounded from above. Our proof follows one given byTrudinger and Wang in the Monge-Amp\`ere case. Thus, like theirs, our argumentis local and yields a quantitative estimate on the $g$-convexity. As a resultour differentiability result is new even in the optimal transport case: weweaken previously required domain convexity conditions. Moreover in the optimaltransport case and the Monge-Amp\`ere case our key assumptions, namely A3w anddomain convexity, are necessary.

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