# Analytic polyhedron

A domain $ \Pi $
of the complex space $ \mathbf C ^ {n} $,
$ n \geq 1 $,
which can be represented by inequalities $ | f _ {i} (z) | < 1 $,
where the functions $ f _ {i} (z) $,
$ i = 1 \dots m $,
are holomorphic in some domain $ D \subset \mathbf C ^ {n} $
containing $ \Pi $,
i.e. $ \Pi = \{ {z \in D } : {| f _ {i} (z) | < 1, i = 1 \dots m } \} $.
It is also assumed that $ \Pi $
is compact in $ D $.
If $ f _ {i} ( z ) $
are polynomials, the analytic polyhedron is said to be a polynomial polyhedron. If $ m = n $
and $ f _ {i} ( z ) = a _ {i} z _ {i} $,
the analytic polyhedron is called a polydisc. The sets $ \sigma _ {i} = \{ {z \in D } : {| f _ {i} ( z) | = 1; | f _ {j} ( z ) | < 1, j \neq i } \} $
are called the faces of the analytic polyhedron. The intersection of any $ k $
different faces $ (2 \leq k \leq n ) $
is said to be an edge of the analytic polyhedron. If $ m \geq n $
and all faces have dimension $ 2n - 1 $,
while no edge has dimension exceeding $ 2n - k $,
the analytic polyhedron is a Weil domain. The set of $ n $-
dimensional edges $ \sigma _ {i _ {1} \dots i _ {n} } = \sigma _ {i _ {1} } \cap \dots \cap \sigma _ {i _ {n} } $
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 $ \Pi $ defined above is sometimes said to be an analytic polyhedron of order $ m $( 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.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Analytic_polyhedron&oldid=45179