  In this paper, we show that the rate of convergence in periodic
homogenization of convex Hamilton-Jacobi equations is always $O(\varepsilon)$,
which is optimal. This is a natural extension of a result concerning stable
norms in metric geometry [4] that is essentially equivalent to the
homogenization of convex static Hamilton-Jacobi equations. Another extremely
interesting question in this direction is whether the $O(\varepsilon)$ rate
holds in the nonconvex setting. We present a special nonconvex example with
$O(\varepsilon)$ convergence rate, which relies on identifying the shape of the
effective Hamiltonian and game theory interpretation formulas.
