Viacheslav Krivokolesko
Integral representation of holomorphic functions in bounded $n$-circular linear convex domain with piecewise regular boundary
In [1] we have obtained a new integral representation for holomorphic functions on bounded linearly convex domains with piecewise regular boundary (linearly convex polyhedra). In this case, the natural question is how can the condition of piecewise regular boundaries of the region be weakened? To answer this question, it seems natural to first consider a simpler situation: the case of $n$-circular linearly convex polyhedra.
Article [2] shows a detail of the integral representation of [1] for this situation. The integrals on the ''faces'', ''edges'', ''tops'' of linear convex polyhedron [1] are reduced to repeated integrals of the projections of the ''faces'', ''edges'', ''tops'' on the Reinhardt and integrals on skeleton of the unit polycylinder. In this study on the linearly convex $n$-circular domain is reduced to the investigation of the projection of this region on a Reinhardt diagram [3].
Note that the right side of the integral representation from [1] is the sum of some terms. In the integration of holomorphic monomials over the boundary of particular area we will get some identities [2], related both to the degree of the monomial, and with the geometrical parameters of the treated area. In [1] the concept of mixed Levian is introduced. For example, mixed Levian of the first order is the determinant of Levy and is associated with the curvature of a surface for which it is located. In the mixed Levians of the second-order the functions involved define the intersection between two surfaces.
So far, we do not know the answer to the question: what is the geometric meaning of mixed Levians of order $k>1$?
REFERENCES
[1] V. Krivokolesko, A. Tsikh, Integral Representations in Linearly Convex Polyhedra, Siberian Mathematical Journal 46 (2005), 579-593.
[2] V. Krivokolesko, Integral Representations for Linearly Convex Polyhedra and Some Combinatorial Identities, Journal Siberian Federal University. Mathematics & Physics 2(2) (2009), 176-188.
[3] M. Forsberg, M. Passare, A. Tsikh, Laurent Determinants and Arrangements of Hyperplane Amoebas, Advances in Mathematics 151 (2000), 45-70.