On Linearly Convex Hartogs Domains in $\mathbb C^2$ with a Fractal Structure

In the 1970s, it was proved that a bounded linearly convex domain with smooth boundary in $\mathbb C^n $ is homeomorphic to an open ball. If the boundary of a bounded linearly convex domain in $\mathbb C^n $ is allowed not to be smooth, then the domain may be of a different topological type. The projection of the complex plane $a_1z_1+\ldots+a_nz_n+c=0$ onto the Hartogs diagram in $\mathbb C^n$ with symmetry plane $z_n=0$ has a simple geometric shape only for $n=2$: in that case, this is a circular cone with vertex in the plane $z_2=0$. This fact allows one to construct linearly convex Hartogs domains in $\mathbb C^2$ with symmetry plane $z_2=0$ whose projections onto the Hartogs diagram have a fractal structure.