Internal characteristics of domains in $\mathbb {C}^{n}$
Volume 111 / 2014
Abstract
This paper is devoted to internal capacity characteristics of a domain $D\subset \mathbb{C}^{n}$, relative to a point $a\in D$, 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 $D\subset \mathbb{C}^{n}$ and its boundary $\partial D$ relative to a point $a\in D$ 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 $\mathbb{C}^{n}$. 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 $\mathbb{Z}^{n}$. Some results are closely related to results on the $s$th 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 $a$.
