Alexandrov's estimate revisited

Alexandrov's estimate states that if Omega is a bounded open convex domain in R-n and u:(Omega) over bar -> R is a convex solution of the Monge-Ampere equation detD(2)u=f that vanishes on partial derivative Omega, then |u(x)-u(y)|<=omega(|x-y|)(integral(Omega)f)(1/n) for omega(delta)= C-n diam(Omega)n-1/n delta(1/n). We establish a variety of improvements of this, depending on the geometry of partial derivative Omega. For example, we show that if the curvature is bounded away from 0, then the estimate remains valid if omega(delta) is replaced by C Omega delta(1/2+1/2n). We determine the sharp constant C-Omega when n=2, and when n >= 3 and partial derivative Omega is C-2, we determine the sharp asymptotics of the optimal modulus of continuity omega Omega(delta) as delta -> 0. For arbitrary convex domains, we characterize the scaling of the optimal modulus omega Omega. Our results imply in particular that unless partial derivative Omega has a flat spot, omega(Omega)(delta)=o(delta(1/n)) as delta -> 0, and under very mild nondegeneracy conditions, they yield the improved Holder estimate, omega(Omega)(delta)<= C delta(alpha) for some alpha>1/n.