A self-similar infinite binary tree is a solution of Steiner problem
Metric Geometry
2023-02-07 v1 Combinatorics
Abstract
We consider a general metric Steiner problem which is of finding a set with minimal length such that is connected, where is a given compact subset of a given complete metric space ; 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
Cite
@article{arxiv.2302.02189,
title = {A self-similar infinite binary tree is a solution of Steiner problem},
author = {D. Cherkashin and Y. Teplitskaya},
journal= {arXiv preprint arXiv:2302.02189},
year = {2023}
}