A Self-Similar Infinite Binary Tree Is a Solution to the Steiner Problem

Fractal and Fractional, vol. 7, pp. 414

Abstract

We consider a general metric Steiner problem, which involves finding a set S with the minimal length, such that S∪A is connected, where A is a given compact subset of a given complete metric space X; a solution is called the Steiner tree. Paolini, Stepanov, and Teplitskaya in 2015 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 exhibits self-similar fractal properties).

Authors 2

1. Danila Dmitrievich Cherkashin corresponding

   Bulgarian Academy of Sciences · Institute of Mathematics and Informatics

   Affiliation as printed

   Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria

2. Yana Teplitskaya Aachen Mathematical Institute

   Centre National de la Recherche Scientifique · Université Paris-Saclay · Leiden University · Laboratoire de Mathématiques d'Orsay

   Affiliation as printed

   Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, Centre National de la Recherche Scientifique (CNRS), 91405 Orsay, France

   Mathematical Institute, Leiden University, 2311 EZ Leiden, The Netherlands