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We study nonparametric estimation of convex regression and density functions by methods of least squares (in the regression and density cases) and maximum likelihood (in the density estimation case). We provide characterizations of these estimators, prove that they are consistent and establish their asymptotic distributions at a fixed point of positive curvature of the functions estimated. The asymptotic distribution theory relies on the existence of an \"invelope function\" for integrated two-sided Brownian motion +t4which is established in a companion paper by Groeneboom, Jongbloed and Wellner.