Mean curvature bounds for the obstacle problem

We prove optimal $(n-1)$-semiconvexity estimates for solutions to the classical obstacle problem, possibly with a smooth right-hand side. We deduce that the mean curvature of the free boundary is universally bounded above in all dimensions. Existing examples show that full curvature bounds fail in dimensions three and higher. Noticeably, these results are novel even for a constant right-hand side in the $n=2$ case.