The *-product of domains in several complex variables

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