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. For any compact subset of $\mathbb{R}^d$, we provide methods for computing upper bounds on these quantities from point cloud data. The bounds converge to the respective quantities as the point cloud becomes dense in the set, and the rate of convergence for the bound on the reach is given under a weak regularity condition. We also introduce the $\beta$-reach, a generalization of the reach that excludes small-scale features of size less than a parameter $\beta\in[0,\infty)$. Numerical studies suggest how the $\beta$-reach can be used in high-dimension to infer the reach and other geometric properties of smooth submanifolds.

翻译：集合的凸性可推广为两个较弱的性质：可达性与$r$-凸性；两者均描述集合边界的正则性。针对$\mathbb{R}^d$中任意紧子集，我们提出了基于点云数据计算这些量上界的方法。随着点云在集合中趋于稠密，所得上界收敛至对应真值，并在弱正则条件下给出了可达性上界的收敛速率。此外，我们引入$\beta$-可达性——该推广通过参数$\beta\in[0,\infty)$滤除尺度小于该参数的细微特征。数值实验表明，$\beta$-可达性可在高维空间中用于推断光滑子流形的可达性及其他几何性质。