In the works of Darboux and Walsh, it was remarked that a one-to-one self-mapping of
R
3
\mathbb {R}^{3}
which sends convex sets to convex ones is affine. It can be remarked also that a
C
2
\mathcal {C}^{2}
-diffeomorphism
F
:
U
→
U
′
F:U\to U^{’}
between two domains in
C
n
\mathbb {C}^{n}
,
n
≥
2
n\ge 2
, which sends pseudoconvex hypersurfaces to pseudoconvex ones is either holomorphic or antiholomorphic.
In this paper we are interested in the self-mappings of
C
n
\mathbb {C}^{n}
which send convex hypersurfaces to pseudoconvex ones. Their characterization is the following: A
C
2
\mathcal {C}^{2}
-diffeomorphism
F
:
U
′
→
U
F:U’\to U
(where
U
′
,
U
⊂
C
n
U’, U\subset \mathbb {C}^{n}
are domains) sends convex hypersurfaces to pseudoconvex ones if and only if the inverse map
Φ
:=
F
−
1
\Phi := F^{-1}
is weakly pluriharmonic, i.e., if it satisfies some nice second order PDE very close to
∂
∂
¯
Φ
=
0
\partial \bar {\partial }\Phi = 0
. In fact all pluriharmonic
Φ
\Phi
’s do satisfy this equation, but there are also other solutions.