  We prove a strong form of the comparison principle for the elliptic
Monge-Ampere equation, with a Dirichlet boundary condition interpreted in the
viscosity sense. This comparison principle is valid when the equation admits a
Lipschitz continuous weak solution. The result is tight, as demonstrated by
examples in which the strong comparison principle fails in the absence of
Lipschitz continuity. This form of comparison principle closes a significant
gap in the convergence analysis of many existing numerical methods for the
Monge-Ampere equation. An important corollary is that any consistent, monotone,
stable approximation of the Dirichlet problem for the Monge-Ampere equation
will converge to the viscosity solution.
