For the Monge-Ampère obstacle problem, any flat piece of the non-strictly convex free boundary has dimension less than (n+q)/2, and this bound is optimal.
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On the singular set of the free boundary for a Monge-Amp\`ere obstacle problem
For the Monge-Ampère obstacle problem, any flat piece of the non-strictly convex free boundary has dimension less than (n+q)/2, and this bound is optimal.