On the Steiner tree connecting a fractal set

Abstract

We construct an example of an infinite planar embedded self-similar binary tree $\hat{I}\pounds$ which is the essentially unique solution to the Steiner problem of finding the shortest connection of a given planar self-similar fractal set $C$ of positive Hausdorff dimension. The set $C$ can be considered the set of leaves, or the ``boundary``, of the tree $\hat{I}\pounds$, so that $\hat{I}\pounds$ is an irreducible solution to the Steiner problem with datum $C$ (i.e. $\hat{I}\pounds \setminus C$ is connected).