Quadratic convexity

We say that a setA ⊂ ℂ ⁿ is quadratically convex if its complement is a union of quadratic hypersurfaces. Some geometric properties of quadratically convex sets are investigated; in particular, they are related to lineally convex sets in a space of higher dimension. We say thatA is strongly quadratically convex if a certain generalization of the Fantappiè transform is surjective, which in effect means that we have a representation for any function holomorphic onA as a superposition of reciprocals of quadratic expressions. The main theorem in this paper gives a sufficient condition for a compact set to be strongly quadratically convex. Using integral representation formulas for holomorphic functions, an explicit inversion formula for the transform is obtained.