The Anisotropic Convexity of Domains and the Boundary Estimate for Two Monge-Ampère Equations

We study the exact effect of the anisotropic convexity of domains on the boundary estimate for two Monge-Ampère Equations: one is singular which is from the proper affine hyperspheres with constant mean curvature; the other is degenerate which is from the Monge-Ampère eigenvalue problem. As a result, we obtain the sharp boundary boundary estimates and the optimal global Hölder regularity for the two equations.