I'm interested in the regularity of solutions to Monge-Ampère equations in a bounded convex domain $\Omega\subset\mathbb{R}^n$. It seems that the following statement can be deduced from known results:



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> **Proposition.** Let $u\in C^0(\overline{\Omega})$ be a convex function satisfying the following conditions:
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> (a) the Monge-Ampère measure of $u$ in $\Omega$ is a positive smooth function times the Lebesgue measure;
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> (b) $u$ has infinite slope at every boundary point (or equivalently, the graph of $u$ does not have non-vertical supporting planes at any boundary point).
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> Then $u$ is smooth in $\Omega$.
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> 


However, since my understanding of the known results is rather limited and superficial, I'm not sure whether this is correct. Here's my argument: First, by Evans-Krylov theory, if $u$ is strictly convex , then condition (a) would imply the smoothness (see e.g. Theorem 3.1 in [this survey of Trudinger-Wang](https://maths-people.anu.edu.au/%7Ewang/publications/MA.pdf)). If $u$ is not strictly convex, then there is a supporting affine function $a$ with $u\geq a$ such that the closed convex set $\{u=a\}$ is not a single point. But it follows from "balancing of sections" that this set cannot have extremal points in $\Omega$, hence must meet $\partial\Omega$ (see e.g. Theorem 7 in [these notes of Connor Mooney](https://www.math.uci.edu/%7Emooneycr/MongeAmpere_Notes.pdf), which violates condition (b). Q.E.D.


So my question is: **Are the above proposition and argument correct? If yes, what are the references to cite if I want to quote the proposition in a paper?**



Regularity of Aleksandrov solution to Monge-Ampère equation with infinite slope at boundary
The proposition is true. One can argue as follows: for $x \in \Omega$, let $L$ be a supporting linear function to $u$ at $x$. Then $L < u$ on $\partial \Omega$ by (b). For $h > 0$ small we conclude that $S\_h := \{u < L + h\}$ is compactly contained in $\Omega$, that $u|\_{\partial S\_h}$ is affine, and that $\det D^2u$ is positive and smooth on $\overline{S\_h}$. By the interior smoothness of solutions to the Dirichlet problem with affine boundary data (first proven by Cheng-Yau using geometric methods in [this paper](https://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.3160300104) and later proven by Lions using PDE techniques in [this paper](https://link.springer.com/article/10.1007/BF01165928)) we conclude that $u$ is smooth.


