all extreme points of $\mathcal{C}$ lie on the boundary $\partial\Omega$. Recall the definition of $\mathcal{C}$ in , and define the set $T:=\{x\in\Omega\,:\,u(x)=\psi(x)\}$, where $\psi$ is the obstacle.

\[l403\] Let $u_0\in\mathcal{S}[\varphi,\psi]$ be the maximizer. The obstacle $\psi$ is a convex function in $\Omega$ satisfying $\psi<\varphi$ on $\partial\Omega$. If all extreme points of $\mathcal{C}$ lie on the boundary $\partial\Omega$, then ${\mbox{dist}}(\mathcal{\overline C},\overline{T})>c_0$ for some positive constant $c_0$.

This follows easily from the convexity.

\[l404\] Assume that $\varphi$ is uniformly convex in a neighborhood of $\Omega$. then $\mathcal{G}_0$ contains no line segments with both endpoints on $\partial\mathcal{G}_0$.

By Lemma \[l403\], we can restrict our discussion on a sub-domain $\Omega'\subset\Omega$ satisfying ${\mbox{dist}}(\Omega',T)>c_0$ and $\{\mbox{extreme points of }\mathcal{C}\}\subset\partial\Omega'\cap\partial\Omega$. Let $u_0$ be the maximizer of $J_\alpha$ and $$\label{e412}
   \bar S[u_0,\Omega']:=\{v\in C(\overline\Omega')\,:\,v \mbox{ convex }, v_{\partial\Omega'}=u_0, N_v(\Omega')\subset N_{u_0}(\overline\Omega')\}.$$ Note that since ${\mbox{dist}}(\Omega',T)>c_0$, when restricting on $\Omega'$, $u_0$ is naturally a maximizer of $J_\alpha$ over $\bar S[u_0,\Omega']$ without obstacle. Therefore, we can apply a similar local approximation in [@TW10] as follows:

> *Claim:* There exists a sequence of smooth, uniformly convex solutions $u_m\in W^{4,p}(\Omega')$ $(\forall p<\infty)$ of $$\label{e413}
>    U^{ij}w_{ij}=f+\beta_m\chi_{D_m}\quad\mbox{in }\Omega'$$ such that $$\label{e414}
>    |u_m-u|\to0\quad\mbox{uniformly in }\Omega',$$ where $D_m=\{x\in\Omega'\,:\,{\mbox{dist}}(x,\partial\Omega')<2^{-m}\}$, $\chi$ is the characteristic function, and $\beta_m>0$ is 