最大距离问题与最小支撑树
The Maximum Distance Problem and Minimal Spanning Trees
摘要：
给定一个紧的$E \subset \mathbb{R}^n$和$s > 0$，最大距离问题寻找最小一维Hausdorff测度的$\mathbb{R}^n$的一个紧的连通子集，其$s$ -邻域覆盖$E$ .对于$E\subset \mathbb{R}^2$，我们证明，最小化的最小生成树连接半径$s$，其中覆盖$E$的球的中心，解决了最大距离问题。证明这个结果的主要困难被引理3.5的证明克服了，它指出人们能够用有限数量的半径为$s$的球覆盖$\mathbb{R}^2$中的Lipschitz曲线$Γ$的$s$ -邻域，并将它们的中心与另一条Lipschitz曲线$Γ_\ast$连接，其中$\mathcal{H}^1(Γ_\ast)$任意接近于$\mathcal{H}^1(Γ)$。我们还提出了一个开放源代码包的计算探索的最大距离问题，使用最小生成树，可在https://github.com/mtdaydream/MDP_MST。
英文摘要：
Given a compact $E \subset \mathbb{R}^n$ and $s > 0$, the maximum distance problem seeks a compact and connected subset of $\mathbb{R}^n$ of smallest one dimensional Hausdorff measure whose $s$-neighborhood covers $E$. For $E\subset \mathbb{R}^2$, we prove that minimizing over minimum spanning trees that connect the centers of balls of radius $s$, which cover $E$, solves the maximum distance problem. The main difficulty in proving this result is overcome by the proof of Lemma 3.5, which states that one is able to cover the $s$-neighborhood of a Lipschitz curve $Γ$ in $\mathbb{R}^2$ with a finite number of balls of radius $s$, and connect their centers with another Lipschitz curve $Γ_\ast$, where $\mathcal{H}^1(Γ_\ast)$ is arbitrarily close to $\mathcal{H}^1(Γ)$. We also present an open source package for computational exploration of the maximum distance problem using minimum spanning trees, available at https://github.com/mtdaydream/MDP_MST.
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