  We show that, in the periodic homogenization of uniformly elliptic
Hamilton-Jacobi equations in any dimension, the effective Hamiltonian does not
necessarily inherit the quasiconvexity property (in the momentum variables) of
the original Hamiltonian. This observation is in sharp contrast with the first
order case, where homogenization is known to preserve quasiconvexity. We also
show that the loss of quasiconvexity is, in a way, generic: when the spatial
dimension is $1$, every convex function $G$ can be modified on an arbitrarily
small open interval so that the new function $\tilde{G}$ is quasiconvex and,
for some $1$-periodic and Lipschitz continuous $V$, the effective Hamiltonian
arising from the homogenization of the uniformly elliptic Hamilton-Jacobi
equation with the Hamiltonian $H(p,x)=\tilde{G}(p)+V(x)$ is not quasiconvex.
