and , $u_0^*$ is strictly convex at $0$, the set $\{u_0^*<h\}$ is strictly contained in $\Omega^*$ provided $h>0$ is small. Note that $u_m^*$ converges to $u_0^*$. By Lemma \[l203\] we have the estimate $${\mbox{det}}\,D^2u_m^*\leq C_1$$ near the origin in $\Omega^*$. Note also that in Lemma \[l203\], the constant $C_1$ depends on $\inf f$ but not on $\sup f$. In other words, the large constant $\beta_m$ in does not affect the bound $C_1$. Therefore, sending $m\to\infty$, we obtained $${\mbox{det}}\,D^2u_0^*\leq C$$ near the origin. This is in contradiction with the assertion that ${\mbox{det}}\,D^2u_0^*$ is not bounded from above near the origin.

Regularity
==========

We can now give the proof of Theorem \[t101\], which is divided into two parts:

$C^{1,\alpha}$ regularity
-------------------------

Assume that $\psi$ is convex and satisfies $\psi<\varphi$ on $\partial\Omega$. Let $u$ be the maximizer of and $\mathcal{G}_u$ be the graph of $u$ over $\Omega$. From Section 4 we know $\mathcal{G}_u$ is strictly convex. The $C^{1,\alpha}$ estimate for strictly convex solutions of Monge-Ampère equations was obtained by Caffarelli [@Caf91]. Here we adopt a similar argument from [@TWma].

For an arbitrary point on $\mathcal{G}_u$, by choosing appropriate coordinates and a rotation in $\mathbb{R}^{n+1}$, we assume it is the origin and $\mathcal{G}_u\subset\{x_3\geq0\}$, and 