On the singular set of the free boundary for a Monge-Ampère obstacle problem

This is a continuation of our earlier work [14] on the Monge-Ampère obstacle problem $$\det D^{2}v=v^q{\chi }_{\{v>0\}},v\ge 0\text{ convex}$$ with $q\in [0,n)$, where we studied the regularity of the strictly convex part of the free boundary. In this work, we examine the non-strictly convex part of the free boundary and establish optimal dimension bounds for its flat portion. Additionally, we investigate the strong maximum principle and a stability property for this Monge-Ampère obstacle problem.