  We prove that solutions to the Monge-Ampere inequality $$\det D^2u \geq 1$$
in $\mathbb{R}^n$ are strictly convex away from a singular set of Hausdorff
$n-1$ dimensional measure zero. Furthermore, we show this is optimal by
constructing solutions to $\det D^2u = 1$ with singular set of Hausdorff
dimension as close as we like to $n-1$. As a consequence we obtain $W^{2,1}$
regularity for the Monge-Ampere equation with bounded right hand side and
unique continuation for the Monge-Ampere equation with sufficiently regular
right hand side.
