The *-product of domains in several complex variables

In this article we continue the research, carried out in \cite{zajac}, on computing $*$-product of domains $\CC^N$. Assuming that $0\in G\subset\CC^N$ is an arbitrary Runge domain and D\subset\CC^N$ a bounded, smooth linearly convex (or non-decreasing union such ones), establish geometric relation between $D*G$ another $\CC^N$ which 'extremal' (in appropriate sense) with respect to special coefficient multiplier dependent only dimension $N$. Next, for $N=2$, derive characterization latter expressed terms planar geometry. These two results, when combined together, give formula allows calculate two-dimensional $D$ $G$ satisfying outlined assumptions.