We prove that in a bounded strictly convex open set
Ω
\Omega
in
R
n
\mathbb {R}^n
, the problem
\[
{
det
∇
2
u
=
f
(
x
)
,
u
|
∂
Ω
=
φ
,
\begin {cases} \det \nabla ^2u=f(x),\ u|_{\partial \Omega }=\varphi , \end {cases}
\]
where
f
>
0
,
f
∈
C
∞
(
Ω
¯
)
,
φ
∈
C
∞
(
∂
Ω
)
f>0,f\in C^\infty (\overline \Omega ), \varphi \in C^\infty (\partial \Omega )
, has a unique strictly convex solution
u
∈
C
∞
(
Ω
¯
)
u\in C^\infty (\overline \Omega )
. This result extends to an arbitrary metric a theorem which has been proved by Caffarelli-Nirenberg-Spruck in the case of the Euclidean metric.