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 $\Gamma$ in $\mathbb{R}^2$ with a finite number of balls of radius $s$, and connect their centers with another Lipschitz curve $\Gamma_\ast$, where $\mathcal{H}^1(\Gamma_\ast)$ is arbitrarily close to $\mathcal{H}^1(\Gamma)$. 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.
翻译:以 $E\ subset\ subbb{R ⁇ }R ⁇ n 和 $ > 0 美元为核心,最大距离问题寻求一个最小型的一维Hausdorff 度量的一维美元和连结子子集$mathb{R ⁇ 2$。对于美元= subset\ mathb{R ⁇ 2$,我们证明最大限度地减少连接半径美元球中心的最小横贯树,这覆盖了美元=E$,解决了最大距离问题。证明这一结果的主要困难在于Lemma 3.5的证明,该证明一个人能够用$\ mathbb{R ⁇ 2$m=Gammas=2$x一定的半径值,并将他们的中心与另一块利普西茨曲线 $\ Gammama_ ast$x, 其中$\ m*1 (\ gamamamam_ast) 来任意地接近于 $\ mathcalH{M\\\\\ mrealalal roadalal roalalalalalal roup sal searmaxal.