论文标题
最大距离最小化问题的概述
An overview of maximal distance minimizers problem
论文作者
论文摘要
考虑一个紧凑型$ M \ subset \ mathbb {r}^d $和$ l> 0 $。最大的距离最小化问题是在最多最大化$ l $的长度（一维Hausdorff Measure $ \ Mathcal H $）的连接的紧凑型集合$σ$中，该$ l $最小化\ [\ max_ {y \ in m} dist（y，σ），\]，其中$ dist $ dist for euclidean distrys euclidean distry。 我们对最大距离最小化器和相关问题的结果进行了调查。另外，我们通过显示最大距离的NP硬度最小化问题，确定其$γ$ - convergence，考虑到惩罚形式并讨论解决方案的独特性，从而填补了一些自然空白。我们结束了开放的问题。
Consider a compact $M \subset \mathbb{R}^d$ and $l > 0$. A maximal distance minimizer problem is to find a connected compact set $Σ$ of the length (one-dimensional Hausdorff measure $\mathcal H$) at most $l$ that minimizes \[ \max_{y \in M} dist (y, Σ), \] where $dist$ stands for the Euclidean distance. We give a survey on the results on the maximal distance minimizers and related problems. Also we fill some natural gaps by showing NP-hardness of the maximal distance minimizing problem, establishing its $Γ$-convergence, considering the penalized form and discussing uniqueness of a solution. We finish with open questions.
