In this talk, we discuss some results in collaboration with Alessio Figalli (ETH Zurich) and Ederson Braga (UFC) about the optimal regularity for semi-convex supersolutions to fully nonlinear PDEs. This extends the Caffarelli Kohn-Nirenberg-Spruck apriori estimates in the 80s. It has also some implications for improving regularity for Sobolev embeddings in the presence of some (general) convexity properties. If time permits, some ideas of the proof and open problems will be discussed.