convexity the graphs of $v_g$ and $u_0$ are bounded by the cone $\mathcal{K}$ and the hyperplane $\mathcal{P}$, where $\mathcal{K}$ has the vertex at $(x_0,u_0(x_0))$ and passes through $(\partial\Omega,u_0|_{\partial\Omega})$, and $\mathcal{P}$ is the support plane of $u_0$ at $x_0$. Then we have $|v_g(x)-u_0(x)|\leq C|x-x_0|$. Hence by the assumption on $G(x, t)$, $$\label{e306}
        \int_{\Omega}G(x,v_g(x))\geq C\int_{\Omega}|x-x_0|^{-n}=\infty.$$ That is, $v_g$ cannot be a maximizer.

Replacing $G$ by $\varepsilon_i G$ for a sequence $\varepsilon_i\to0$, accordingly there exists a sequence of maximizers $v_{\varepsilon_i}$ to . Since $u_0$ is itself a maximizer, we have $v_{\varepsilon_i}\to u_0$ as $\varepsilon_i\to 0$ by the concavity of the functional $J_\alpha$. Hence, the sequence $u_i$ can be chosen from $v_{\varepsilon_i}$.

If $u_i$ is smooth, it satisfies the equation $$\label{e307}
        L[u]=f+g_i \quad\mbox{in }\Omega,$$ where $L$ is the operator in , and $g_i=\frac{\partial}{\partial t}G_i(x,t)$ at $t=u_i(x)$. In the later proof of strict convexity, we will need the upper bound estimate for the determinant of $D^2u_0$ which depends on $\sup f$. Since $g_i<0$ in the above approximation, the estimate in Section 2 still applies when turning to the sequence $u_i$.

When studying the strict convexity of enclosed convex hypersurfaces with maximal affine area, one can assume $u_0$ is equal to a linear function $\ell$ on 