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

arXiv (Cornell University)

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).

Authors 2

1. Danila Dmitrievich Cherkashin

   Bulgarian Academy of Sciences · Institute of Mathematics and Informatics

   Affiliation as printed

   Education and Science of Bulgaria , Scientific Programme "Enhancing the Research Capacity in Mathematical Sciences

   Institute of Mathematics and Informatics , Bulgarian Academy of Sciences. The

2. Y. Teplitskaya Aachen Mathematical Institute

   Leiden University

   Affiliation as printed

   Mathematical Institute , Leiden University , Netherlands