关于Steiner问题的唯一性
On uniqueness in Steiner problem
摘要：
我们证明了一组$n$点配置的平面斯坦纳问题的解决方案是不唯一的豪斯多夫维数最多$2n-1$（作为一个子集的$\mathbb{R}^{2n}$）。此外，我们还证明了至少有两个局部极小树具有相同长度的$n$ -点配置集的豪斯多夫维数也至多为$2n-1$。我们使用的方法基本上需要依赖于~ \cite{bierstone1988semianalytic}中发展的次解析集理论。出于这种方法，我们开发了一个通用的设置为类似的问题的唯一性的Steiner树的欧氏平面被替换为任意解析黎曼流形$M$。在这种设置中，我们认为，一组配置拥有两个本地最小的树相同的长度，要么有尺寸$n\dim M-1$或有一个非空的内部。我们提供了一个二维曲面的例子，最后一种选择适用。除上述结果外，我们还研究了在$\mathbb{R}^d$中Steiner问题存在唯一解的$n$ -点组态的集合。我们证明了这个集合是路径连通的。
英文摘要：
We prove that the set of $n$-point configurations for which the solution of the planar Steiner problem is not unique has the Hausdorff dimension at most $2n-1$ (as a subset of $\mathbb{R}^{2n}$). Moreover, we show that the Hausdorff dimension of the set of $n$-point configurations on which at least two locally minimal trees have the same length is also at most $2n-1$. Methods we use essentially require rely upon the theory of subanalytic sets developed in~\cite{bierstone1988semianalytic}. Motivated by this approach we develop a general setup for the similar problem of uniqueness of the Steiner tree where the Euclidean plane is replace by an arbitrary analytic Riemannian manifold $M$. In this setup we argue that the set of configurations possessing two locally-minimal trees of the same length either has the dimension $n\dim M-1$ or has a non-empty interior. We provide an example of a two-dimensional surface for which the last alternative holds. In addition to abovementioned results, we study the set of set of $n$-point configurations for which there is a unique solution of the Steiner problem in $\mathbb{R}^d$. We show that this set is path-connected.
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。
