Sample maximum and minimum

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. 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 $2n-k$.

See also: the Behnke–Stein theorem.

References

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• Lars Hörmander. An Introduction to Complex Analysis in Several Variables, North-Holland Publishing Company, New York, New York, 1973.