自相似无限二叉树是Steiner问题的一种解法
A self-similar infinite binary tree is a solution of Steiner problem
摘要：
考虑一般度量Steiner问题，即寻找一个长度最小的集合$\mathcal{S}$，使得$\mathcal{S} \cup A$连通，其中$A$是给定完全度量空间$X$的给定紧致子集，其解称为Steiner树. Paolini、Stepanov和Teplitskaya提供了一个平面斯坦纳树的例子，它有无限多个分支点连接着一个不可数的点集。我们证明了这样的集合可以有一个正的豪斯多夫维数，这是一个公开的问题（相应的树是一个自相似分形）。
英文摘要：
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).
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
