Alexandrov's estimate revisited

Alexandrov’s estimate states that if $\mathrm{\Omega }$ is a bounded open convex domain in ${\mathbb{R}}^n$ and $u:\bar{\mathrm{\Omega }}\to \mathbb{R}$ is a convex solution of the Monge-Ampere equation $\det D^{2}u=f$ that vanishes on $\mathrm{\partial }\mathrm{\Omega }$, then $$\mathrm{\mid }u(x)-u(y)\mathrm{\mid }\le \omega (\mathrm{\mid }x-y\mathrm{\mid })({\int }_{\mathrm{\Omega }}f{)}^{1\mathrm{/}n}for\omega (\delta )=C_{n}diam(\mathrm{\Omega }{)}^{\frac{n-1}{n}}{\delta }^{1\mathrm{/}n}\mathrm{.}$$ We establish a variety of improvements of this, depending on the geometry of $\mathrm{\partial }\mathrm{\Omega }$. For example, we show that if the curvature is bounded away from $0$, then the estimate remains valid if $\omega (\delta )$ is replaced by $C_{\mathrm{\Omega }}{\delta }^{\frac{1}{2}+\frac{1}{2n}}$. We determine the sharp constant $C_{\mathrm{\Omega }}$ when $n=2$, and when $n\ge 3$ and $\mathrm{\partial }\mathrm{\Omega }$ is $C^2$, we determine the sharp asymptotics of the optimal modulus of continuity ${\omega }_{\mathrm{\Omega }}(\delta )$ as $\delta \to 0$. For arbitrary convex domains, we characterize the scaling of the optimal modulus ${\omega }_{\mathrm{\Omega }}$. Under very mild nondegeneracy conditions, our results yield the improved Holder estimate, ${\omega }_{\mathrm{\Omega }}(\delta )\le C{\delta }^{\alpha }$ for some $\alpha >1\mathrm{/}n$.