new coordinates, $v_k$ converges locally uniformly to $v$. Let $\mathcal{\tilde C}=\{(z_1,z_2),\:\,v(z_1,z_2)=0\}$, and $$\label{e409}
        L=\{(z_1,z_2,0)\,:\,(z_1,z_2)\in\mathcal{\tilde C}\}$$ in $z$-coordinates. $L$ could be a single point (Case I) or a segment on $z_2$-axis (Case II).

*Case I*: In this case, $v$ is strictly convex at $(0,0)$. The strict convexity implies that $Dv$ is bounded on the sub-level set $S_{h,v}(0)$ for small $h>0$. Hence, by locally uniform convergence, $Dv_k$ are uniformly bounded on $S_{h/2,v_k}(0)$. By Lemma \[l204\], we have the determinant estimate $$\label{e410}
        {\mbox{det}}\,D^2v_k\leq C$$ near the origin, where the constant $C$ is uniform with respect to $k$. By the weak continuity of Monge-Ampère measure, $\mu[v]\leq C$ near the origin. The contradiction follows by Lemma 4.1.

*Case II*: In this case, $L$ is a segment, we may also assume that $0$ is an end point of $L$, i.e., $$\mathcal{\tilde C}=\{(0, z_2)\,:\,-1\leq z_2\leq 0\}.$$ Define the linear function $$\label{e411}
        \ell_\varepsilon(z)=\delta_\varepsilon z_2+\varepsilon$$ and $\omega_\varepsilon=\{z\,:\,v(z)\leq\ell_\varepsilon\}$, where $\delta_\varepsilon, \varepsilon$ are chosen such that $\varepsilon\delta_\varepsilon^{-1}\to0$ as $\varepsilon\to0$. By taking the similar transformations and normalizations as in , with respect to $z_2$ direction, one can reduce Case II to Case I. The proof is then finished.

Strict convexity II
-------------------

Next, we rule out the possibility of case (b) that 