# Analytic polyhedron

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In mathematics, especially several complex variables, an analytic polyhedron is a subset of the complex space $\mathbf{C}^n$ of the form

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

where $D$ is a bounded connected open subset of $\mathbf{C}^n$ and $f_j$ are holomorphic on D.[1] If $f_j$ above are polynomials, then the set is called a polynomial polyhedron. Every analytic polyhedron is a domain of holomorphy (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 $2n-k$.[2]

See also: the Behnke–Stein theorem.

## References

1. ^ http://www.emis.de/journals/UIAM/PDF/45-139-145.pdf
2. ^ E. M. Chirka, A. G. Vitushkin (1997). Introduction to Complex Analysis. Springer. pp. 35–36. ISBN 9783540630050.
• Lars Hörmander. An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Company, New York, New York, 1973.