论文标题
关于欧几里得空间中最大距离最小化器的规律性
On regularity of maximal distance minimizers in Euclidean Space
论文作者
论文摘要
我们研究集合$σ$的属性，这是最大距离最小化问题的解决方案，即具有最小长度（一维的Hausdorff度量）的集合的封闭式连接组的集合集合$σ\ subset \ subset \ mathbb {r}^n $满足不足的不足\ [max_} yq y s y]对于给定的紧凑型集合$ M \ subset \ mathbb {r}^n $和一些给定的$ r> 0 $。这样的集合可以被视为最短的网络，即以$ r $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ R $的距离到达每个客户的辐射Wi-Fi电缆。 在本文中，我们证明，任何最大距离最小化$σ\ subset \ mathbb {r}^n $在每个点上最多都有$ 3 $切线射线，而同一点的任何两个切线射线之间的角度至少为$2π/3 $。此外，在平面（对于$ n = 2 $）中，我们表明，带有三个切线射线的点数是有限的，每个最大距离最小化器都是一个有限的简单曲线结合，并与相应侧连续连续的单侧切线。 所有结果均证明了局部最小化器的更一般类别，即在其任意点的社区扰动下最佳的集合。
We study the properties of sets $Σ$ which are the solutions of the maximal distance minimizer problem, i.e. of sets having the minimal length (one-dimensional Hausdorff measure) over the class of closed connected sets $Σ\subset \mathbb{R}^n$ satisfying the inequality \[ max_{y \in M} dist(y,Σ) \leq r \] for a given compact set $M \subset \mathbb{R}^n$ and some given $r > 0$. Such sets can be considered as the shortest networks of radiating Wi-Fi cables arriving to each customer (for the set $M$ of customers) at a distance at most $r$. In this paper we prove that any maximal distance minimizer $Σ\subset \mathbb{R}^n$ has at most $3$ tangent rays at each point and the angle between any two tangent rays at the same point is at least $2π/3$. Moreover, in the plane (for $n=2$) we show that the number of points with three tangent rays is finite and every maximal distance minimizer is a finite union of simple curves with one-sided tangents continuous from the corresponding side. All the results are proved for the more general class of local minimizers, i.e. sets which are optimal under a perturbation of a neighbourhood of their arbitrary point.
