Boundary H\"older gradient estimates for the Monge-Amp\`ere equation

We investigate global H"older gradient estimates for solutions to the Monge-Amp`ere equation $$\mathrm{d}\mathrm{e}\mathrm{t}{D}^{2}u=f\mathrm{i}\mathrm{n}\mathrm{\Omega },$$ where the right-hand side $f$ is bounded away from $0$ and $\mathrm{\infty }$. We consider two main situations when a) the domain $\mathrm{\Omega }$ is uniformly convex and b) $\mathrm{\Omega }$ is flat.