  We construct a counterexample to $W^{2,1}$ regularity for convex solutions to
$$\det D^2u \leq 1, \quad u|_{\partial \Omega} = 0$$ in two dimensions. We also
prove a result on the propagation of singularities in two dimensions that are
logarithmically slower than Lipschitz. This generalizes a classical result of
Alexandrov and is optimal by example.
