Alexandrov’s estimate revisited

Abstract Alexandrov’s estimate states that if $\Omega $ is a bounded open convex domain in $\mathbb {R}^n$ and $u:\bar \Omega \to \mathbb {R}$ is a convex solution of the Monge-Ampère equation $\det D^2 u = f$ that vanishes on $\partial \Omega $ , then $$\begin{align*}|u(x) - u(y)| \le \omega(|x-y|)(\int_\Omega f)^{1/n} \qquad \text{for }\omega(\delta) = C_n\,\text{diam}(\Omega)^{\frac{n-1}n} \delta^{1/n}. \end{align*}$$ We establish a variety of improvements of this, depending on the geometry of $\partial \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 $C_\Omega \delta ^{\frac 12 + \frac 1{2n}}$ . We determine the sharp constant $C_\Omega $ when $n=2$ , and when $n\ge 3$ and $\partial \Omega $ is $C^2$ , we determine the sharp asymptotics of the optimal modulus of continuity $\omega _\Omega (\delta )$ as $\delta \to 0$ . For arbitrary convex domains, we characterize the scaling of the optimal modulus $\omega _\Omega $ . Our results imply in particular that unless $\partial \Omega $ has a flat spot, $\omega _\Omega (\delta ) = o(\delta ^{1/n})$ as $\delta \to 0$ , and under very mild nondegeneracy conditions, they yield the improved Hölder estimate, $\omega _\Omega (\delta ) \le C \delta ^\alpha