1. A Self-Similar Infinite Binary Tree Is a Solution to the Steiner Problem.
- Author
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Cherkashin, Danila and Teplitskaya, Yana
- Subjects
- *
FRACTAL dimensions , *METRIC spaces , *COMMERCIAL space ventures , *TREES , *POINT set theory , *STEINER systems , *METRIC geometry - 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). [ABSTRACT FROM AUTHOR]
- Published
- 2023
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