  This article is about the convex solution $u$ of the Monge--Amp\`ere equation
on an at least 2-dimensional open bounded convex domain with Dirichlet boundary
data and nonnegative bounded right-hand side.
  For convex functions with zero boundary data, an Alexandrov maximum principle
$|u(x)| \leq C \operatorname{dist}(x,\partial\Omega)^\alpha$ is equivalent to
(uniform) H\"older continuity with the same constant and exponent. Convex
$\alpha$-H\"older continuous functions are $W^{1,p}$ for $p < 1/(1{-}\alpha)$.
We prove H\"older continuity with the exponent $\alpha=2/n$ for $n \geq 3$ and
any $\alpha \in (0,1)$ for $n=2$, provided that the boundary data satisfy this
H\"older continuity, and show that these bounds for the exponent are sharp. The
only means is to bound the Hessian determinant of a certain explicit function
on an $n$-dimensional cylinder and to use the comparison princple.
