# Weil domain

A special case of an analytic polyhedron. A bounded domain $D$ in $n$- dimensional space $\mathbf C ^ {n}$ is said to be a Weil domain if there exist $N \geq n$ functions $f _ {i} ( z)$, $i= 1 \dots N$, holomorphic in a fixed neighbourhood $U ( \overline{D}\; )$ of the closure $\overline{D}\;$, such that

1) $D= \{ {z } : {| f _ {i} ( z) | < 1, i = 1 \dots N, z \in U ( \overline{D}\; ) } \}$;

2) the faces of the Weil domain $D$, i.e. the sets

$$\sigma _ {i} = \{ {z \in D } : { | f _ {i} ( z) | = 1 ,\ | f _ {j} ( z) | \leq 1 ,\ j \neq i } \} ,$$

have dimension $2n - 1$;

3) the edges of the Weil domain $D$, i.e. the intersections of any $k$( $2 \leq k \leq n$) different faces, have dimension $\leq 2n - k$.

The totality of all $n$- dimensional edges of a Weil domain is called the skeleton of the domain. The Bergman–Weil representation applies to Weil domains. These domains are named for A. Weil [1], who obtained the first important results for these domains.

#### References

 [1] A. Weil, "L'intégrale de Cauchy et les fonctions de plusieurs variables" Math. Ann. , 111 (1935) pp. 178–182 [2] B.V. Shabat, "Introduction of complex analysis" , 1–2 , Moscow (1976) (In Russian) [3] V.S. Vladimirov, "Methods of the theory of functions of several complex variables" , M.I.T. (1966) (Translated from Russian)