Calculus of Variations and Geometric Measure Theory

E. Paolini - E. Stepanov

On the Steiner tree connecting a fractal set

created by paolini on 04 Apr 2023
modified on 18 May 2023



Inserted: 4 apr 2023
Last Updated: 18 may 2023

Year: 2023

ArXiv: 2304.01932 PDF


We construct an example of an infinite planar embedded self-similar binary tree $\Sigma$ 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 $\Sigma$, so that $\Sigma$ is an irreducible solution to the Steiner problem with datum $C$ (i.e. $\Sigma\setminus C$ is connected).

Keywords: Steiner problem, Steiner tree, fractal set