A priori estimates for solutions of fully nonlinear equations with convex level set

We derive an a priori C-2,C-alpha estimate for solutions of the fully non-l inear elliptic equation F(D(2)u) = 0, provided the level set Sigma = {M \ F (M) = 0} satisfies: (a) Sigma boolean AND {M \ TrM = t} is strictly convex for ail constants t; (b) the angle between the identity matrix I and the no rmal F-ij to Sigma is strictly positive on the non-convex part of Sigma. Mo reover, we do not need any convexity assumption on F in the course of the p roof for the two dimensional case, as the classical result indicates.