The convexity of a set can be generalized to the two weaker notions of reach and $r$-convexity; both describe the regularity of a set's boundary. In this article, these two notions are shown to be equivalent for closed subsets of $\mathbb{R}^d$ with $C^1$ smooth, $(d-1)$-dimensional boundary. In the general case, for closed subsets of $\mathbb{R}^d$, we detail a new characterization of the reach in terms of the distance-to-set function applied to midpoints of pairs of points in the set. For compact subsets of $\mathbb{R}^d$, we provide methods of approximating the reach and $r$-convexity based on high-dimensional point cloud data. These methods are intuitive and highly tractable, and produce upper bounds that converge to the respective quantities as the density of the point cloud is increased. Simulation studies suggest that the rates at which the approximation methods converge correspond to those established theoretically.

翻译：一组的共性可以普遍化为两个较弱的“ 到达” 和“ 美元” 概念; 两者都描述一组边界的规律性。 在本条中, 这两个概念被显示为相当于 $\ mathbb{R ⁇ d$的封闭子集, 平滑, $(d-1)$(d-1)$- 维边界。 在一般情况下, 对于 $\mathbb{R ⁇ d$ 的封闭子集, 我们详细描述对一组点中点适用的远到定函数的覆盖范围的新特征。 对于 $\ mathbb{R ⁇ d$ 的紧凑子集, 我们提供基于高维点云数据的接近和 $r$- convexity 的方法。 这些方法不直观, 并且高度可移动, 并产生随着点云密度的增加而与相应数量相趋近的上限。 模拟研究表明, 近似方法汇合率的速率与已确定的理论性相符。