# Analytic polyhedron

A domain of the complex space , , which can be represented by inequalities , where the functions , , are holomorphic in some domain containing , i.e. . It is also assumed that is compact in . If are polynomials, the analytic polyhedron is said to be a polynomial polyhedron. If and , the analytic polyhedron is called a polydisc. The sets are called the faces of the analytic polyhedron. The intersection of any different faces is said to be an edge of the analytic polyhedron. If and all faces have dimension , while no edge has dimension exceeding , the analytic polyhedron is a Weil domain. The set of -dimensional edges forms the skeleton of the analytic polyhedron. The concept of an analytic polyhedron is important in problems of integral representations of analytic functions of several variables.

#### References

[1] | B.V. Shabat, "Introduction of complex analysis" , 2 , Moscow (1976) (In Russian) |

#### Comments

The analytic polyhedron defined above is sometimes said to be an analytic polyhedron of order (cf. [a1]).

#### References

[a1] | L. Hörmander, "An introduction to complex analysis in several variables" , North-Holland (1973) pp. Chapt. 2.4 |

**How to Cite This Entry:**

Analytic polyhedron. E.D. Solomentsev (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Analytic_polyhedron&oldid=12335