Internal characteristics of domains in Cn

Abstract

This paper is devoted to internal capacity characteristics of a domain D subset of C-n, relative to a point a is an element of D, which have their origin in the notion of the conformal radius of a simply Connected plane domain relative to a point. Our main goal is to study the internal Chebyshev constants and transfinite diameters for a domain D subset of C-n and its boundary partial derivative D relative to a point a is an element of D in the spirit of the author's article [Math. USSR-Sb. 25 (1975), 350-364], where similar characteristics have been investigated for compact sets in C-n. The central notion of directional Chebyshev constants is based on the asymptotic behavior of extremal monic polynomials and copolynomials in directions determined by the arithmetic of the index set Z(n). Some results are closely related to results on the sth Reiffen pseudometrics and internal directional analytic capacities of higher order (Jarnicki-Pflug, Nivoche) describing the asymptotic behavior of extremal copolynomials in varied directions when approaching the point a.