Bernstein theorem and regularity for a class of monge-ampère equations

In this paper we first introduce a transform for convex functions and use it to prove a Bernstein theorem for a Monge-Ampère equation in half space.We then prove the optimal global regularity for a class of Monge-Ampère type equations arising in a number of geometric problems such as Poincaré metrics, hyperbolic affine spheres, and Minkowski type problems.