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arXiv · 2304.01932

On the Steiner tree connecting a fractal set

Emanuele Paolini·Eugene Stepanov
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Abstract

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).

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Emanuele Paolini, Eugene Stepanov. 2023-04-04. On the Steiner tree connecting a fractal set. https://arxiv.org/abs/2304.01932

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