The Distance Between the Perturbation of a Convex Function and its $\Gamma$-regularization

In the study of a non-convex minimization problem by Lachand-Robert and Peletier, they found that the difference between the compactly supported perturbation $u+\epsilon h$ of a strictly convex function $u$, and the $\Gamma$-regularization of $u+\epsilon h$, is at most $o(\epsilon)$. Here we find that this result is optimal, albeit they expected a much stronger estimate.