A self-similar infinite binary tree is a solution of Steiner problem
Abstract
We consider a general metric Steiner problem which is of finding a set S with 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 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).
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