Analytic polyhedron

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


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

where $D$ is a bounded connected open subset of $C^{n}$, $$f_j$$ are holomorphic on $D$ and $P$ is assumed to be relatively compact in $D$. 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 contained in 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 any $k$ of the above hypersurfaces has dimension no greater than $2n-k$.