The Integral Representation for Polyhedral Domains of ℂn

Abstract

By a polyhedral domain in ℂn we mean a bounded domain D in ℂn which satisfies the following two conditions. 1) The boundary ∂D of D is homeomorphic to the polyhedron \( \dot S^{{\text{2n}}} \) of the boundary complex \( \dot S^{2n} \) of the standard 2n-dimensional simplex S2n of the 2n-dimensional Euclidean space. We denote by \( \sigma _{\text{j}_\text{l} \cdot \cdot \text{j}_\text{k} } \) a subset of ∂D which corresponds to a (2n−k)-dimensional simplex S(2n−k) of \( \dot S^{2n} \) under the homeomorphism. 2) For some differentiable functions \( w_j = w_j (z) = u_j (x,y) + \sqrt { - 1} \,v_j (x,y) \) on some bounded domain R in ℂn and for some bounded domain Rj in the wj-plane with piecewise smooth boundary (j = 1,…,N), D is a component of \( \{ z\,\varepsilon \,R\mid w_j (z)\,\varepsilon R_j ,\,j = 1, \cdots ,N\} \) such that the topological closure \( \rm \overline D \) of D is contained in R. Without loss of generality we can assume that all the domains Rj are needed to define D.