On convex to pseudoconvex mappings
S. Ivashkovich · Proceedings of the American Mathematical Society · 2009
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