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Quadratic convexity

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Abstract

We say that a setA ⊂ ℂn 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.

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Authors and Affiliations

  1. Department of Physics and Mathematics, Mid-Sweden University, S-851 70, Sundsvall, Sweden

    Jörgen Boo

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  1. Jörgen Boo
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Correspondence to Jörgen Boo.

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Boo, J. Quadratic convexity. J. Anal. Math. 76, 45–65 (1998). https://doi.org/10.1007/BF02786929

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  • DOI: https://doi.org/10.1007/BF02786929

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