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 $$\mathrm{det}\;D^2 u=f\quad\mathrm{in}\;\Omega,$$ where the right-hand side $f$ is bounded away from $0$ and $\infty$. We consider two main situations when a) the domain $\Omega$ is uniformly convex and b) $\Omega$ is flat.