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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 We establish a variety of improvements of this, depending on the geometry of 
$$\begin{align*}|u(x) - u(y)| \le \omega(|x-y|)(\int_\Omega f)^{1/n} \qquad \text{for }\omega(\delta) = C_n\,\text{diam}(\Omega)^{\frac{n-1}n} \delta^{1/n}. \end{align*}$$
$\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 $ for some 
$\alpha>1/n$.
