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Abstract: In this talk, we discuss some sharp regularity results related to supersolutions to fully nonlinear equations that enjoy some convexity property. If time allows, some recent and ongoing results will be discussed. This is joint work with Alessio Figalli (ETH-Zurich) and J. Ederson Braga (UFC) and more recently with Prof. Edgard Pimentel (University of Coimbra, Portugal).