Global $$C^{1,\alpha }$$ C 1 , α Regularity for Monge–Ampère Equation and Convex Envelope

In this paper we establish the global $$C^{1,\alpha }$$ regularity for solutions to the Dirichlet problem of the Monge–Ampère equation $$\det D^2 u=f$$ . By examples we show that our conditions are optimal. Our proof allows the degenerate case $$f\ge 0$$ , including the special case $$f\equiv 0$$ . We also prove the global $$C^{1,\alpha }$$ regularity for the convex envelope of a given function under optimal conditions.