论文标题
在平面凸形封闭平滑曲线的最大距离功能的最小化中
On minimizers of the maximal distance functional for a planar convex closed smooth curve
论文作者
论文摘要
修复一个紧凑型$ M \ subset \ Mathbb {r}^2 $和$ r> 0 $。最大距离功能的最小化器是最小长度的连接集合$σ$，因此\ [max_ {y \ in m} dist（y，σ）\ leq r。 \]查找最大距离最小化器的问题已连接到Steiner树问题。 在本文中，我们考虑了凸的封闭曲线$ m $的情况，其曲率半径大于$ r $（这意味着$ m $是光滑的）。第一部分专门用于$σ$的结构的语句：我们表明，$ b_r（m）\capσ$的任意连接组件的关闭是一个本地施泰纳树，它连接不超过五个顶点。 在第二部分中，我们“在图片中得出”。假设$ y \ in m $的左和右邻域包含在不同点的$ r $ neighborhoods中，$ x_1 $，$ x_2 \ inσ$。我们在$ x_1 $和$ x_2 $的社区中的行为上写下条件，并在假设下通过$ m $将$ y $移动。
Fix a compact $M \subset \mathbb{R}^2$ and $r>0$. A minimizer of the maximal distance functional is a connected set $Σ$ of the minimal length, such that \[ max_{y \in M} dist(y,Σ) \leq r. \] The problem of finding maximal distance minimizers is connected to the Steiner tree problem. In this paper we consider the case of a convex closed curve $M$, with the minimal radius of curvature greater than $r$ (it implies that $M$ is smooth). The first part is devoted to statements on structure of $Σ$: we show that the closure of an arbitrary connected component of $B_r(M) \cap Σ$ is a local Steiner tree which connects no more than five vertices. In the second part we "derive in the picture". Assume that the left and right neighborhoods of $y \in M$ are contained in $r$-neighborhoods of different points $x_1$, $x_2 \in Σ$. We write conditions on the behavior of $Σ$ in the neighborhoods of $x_1$ and $x_2$ under the assumption by moving $y$ along $M$.
