Analytic polyhedron

In mathematics, especially several complex variables, an analytic polyhedron is a subset of the complex space Cⁿ of the form

$\{ z \in D : |f_j(z)| < 1, 1 \le j \le N \}\,$

where D is a bounded connected open subset of Cⁿ and $f_j$ are holomorphic on D. If $f_j$ above are polynomials, then the set is called a polynomial polyhedron. Every analytic polyhedron is a domain of holomorphy and it is thus pseudo-convex.

The boundary of an analytic polyhedron is the union of the set of hypersurfaces

$\sigma_j = \{ z \in D : |f_j(z)| = 1 \}, 1 \le j \le N.$

An analytic polyhedron is a Weil polyhedron, or Weil domain if the intersection of k hypersurfaces has dimension no greater than 2n-k.

See also

• Behnke–Stein theorem
• Bergman–Weil formula

References

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• (also available as ISBN 0-387-17004-9).
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• . Notes from a course held by Francesco Severi at the Istituto Nazionale di Alta Matematica (which at present bears his name), containing appendices of Enzo Martinelli, Giovanni Battista Rizza and Mario Benedicty. An English translation of the title reads as:-"Lectures on analytic functions of several complex variables – Lectured in 1956–57 at the Istituto Nazionale di Alta Matematica in Rome".