Calculus of Variations and Geometric Measure Theory

D. Cherkashin - Y. Teplitskaya

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

created by teplitskaya1 on 10 Feb 2023
modified by cherkashin on 12 Feb 2024



Inserted: 10 feb 2023
Last Updated: 12 feb 2024

Journal: Fractal and Fractional
Volume: 7
Number: 5
Pages: 414
Year: 2023

ArXiv: 2302.02189 PDF


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).

Keywords: Steiner problem, Steiner tree, fractal set, explicit solution