A self-similar infinite binary tree is a solution of Steiner problem

TL;DR

This paper demonstrates that a self-similar infinite binary tree, serving as a Steiner tree connecting an uncountable set, can possess a positive Hausdorff dimension, revealing new fractal properties in geometric optimization.

Contribution

It proves that certain infinite binary trees solving the Steiner problem can have positive Hausdorff dimension, answering an open question about their fractal nature.

Findings

Existence of Steiner trees with positive Hausdorff dimension

Construction of a self-similar fractal Steiner tree

Extension of Steiner problem solutions to fractal sets

Abstract

We consider a general metric Steiner problem which is of finding a set $\mathcal{S}$ with minimal length such that $\mathcal{S} \cup A$ is connected, where $A$ is a given compact subset of a given complete metric space $X$; a solution is called Steiner tree. Paolini, Stepanov and Teplitskaya provided an example of a planar Steiner tree with an infinite number of branching points connecting an uncountable set of points. We prove that such a set can have a positive Hausdorff dimension which was an open question (the corresponding tree is a self-similar fractal).