Global Lipschitz and Sobolev estimates for the Monge-Ampère eigenfunctions of general bounded convex domains

We show that the Monge-Ampère eigenfunctions of general bounded convex domains are globally Lipschitz. The same result holds for convex solutions to degenerate Monge-Ampère equations of the form $\det D^{2}u=M\mathrm{\mid }u{\mathrm{\mid }}^p$ with zero boundary condition on general bounded convex domains in ${\mathbb{R}}^n$ within the sharp threshold $p>n-2$. As a consequence, we obtain global $W^{2,1}$ estimates for these solutions.