There is an extensive literature on the local regularity theory for the real Monge-Ampère equation, largely due to the Caffarelli school. The author thanks C. Mooney for bringing some of these results to his attention. All results surveyed here can be found in [35].

Any convex function on an open set $v:\Omega\subset\mathbb{R}^{n}\to\mathbb{R}$ has an associated Borel measure called the Monge-Ampère measure, defined by

$$MA_{\mathbb{R}}(v)(E)=|\partial v(E)|,$$

where $|\partial v(E)|$ denotes the Lebesgue measure of the image of the subgradient map on $E\subset\Omega$. Given a Borel measure $\mu$, a solution to $MA(v)=\mu$ is called an Aleksandrov solution to $\det(D^{2}v)=\mu;$ if $v\in C^{2}$, this is the classical real Monge-Ampère equation. We shall assume a two-sided density bound

$$\det(D^{2}v)=f\text{ in }B_{1},\quad 0<\Lambda_{1}\leq f\leq\Lambda_{2}.$$

Let $B_{1}\setminus\Sigma$ be the set of strictly convex points of $v$, namely there is a supporting hyperplane touching the graph of $v$ only at one point. Then Caffarelli [8][9][10] shows

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  If $f\in C^{\gamma}(B_{1})$, then $v\in C^{2,\gamma}_{loc}(B_{1}\setminus\Sigma)$. Then by Schauder theory, if $f$ is smooth, then $v$ is smooth in $B_{1}\setminus\Sigma$.

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  If $L$ is a supporting affine linear function to $v$, such that the convex set $\{v=L\}$ is not a point. Then $\{v=L\}$ has no extremal point in the interior of $B_{1}$.

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  The above affine linear set $\{v=L\}$ has dimension $k<n/2$.

Mooney [35] shows further that

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  The singular set $\Sigma$ has $(n-1)$-Hausdorff measure zero. Consequently $B_{1}\setminus\Sigma$ is path connected (because a generic path joining two given points does not intersect a subset of zero $(n-1)$-Hausdorff measure).

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  The solution $v\in W^{2,1}_{loc}(B_{1})$ even if $\Sigma$ is nonempty.