Degenerate Monge-Ampere equations of the type \(\text{det} \, D^2 u = f\) in \(\Omega\), \( f \sim d_{\partial \Omega}^a\) near the boundary \(\partial \Omega\) are considered, where \(D^2 u = [ u_{x_ix_j}]_{i,j=1}^n\), \(\Omega\) is a convex domain, \(d_{\partial \Omega}\) is the distance to the boundary, and \(a > 0\) is a positive power. The main results of the paper are the proof of a localization theorem and of a pointwise \(C^2\) estimate for solutions \(u\). Two estimates of Pogorelov type for solutions of some Monge-Ampere equations are obtained which are ingredients of the proof of the localization theorem. The proof of the \(C^2\) estimate follows from a Liouville type theorem, for which a proof is given in the paper.