Computable Bounds for the Reach and r-Convexity of Subsets of $${{\mathbb {R}}}^d$$

Pub Date : 2024-01-27 DOI:10.1007/s00454-023-00624-8
Ryan Cotsakis
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Abstract

The convexity of a set can be generalized to the two weaker notions of positive 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 sampling scale of the point cloud decreases, 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.

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$${{mathbb {R}}^d$$ 子集的可达性和 r-凸性的可计算边界
一个集合的凸性可以概括为两个较弱的概念:正凸性和r-凸性;这两个概念都描述了一个集合边界的规则性。对于 \({{\mathbb {R}}^d\) 的任意紧凑子集,我们提供了从点云数据计算这些量的上界的方法。随着点云采样尺度的减小,上界会收敛到相应的量,并且在弱正则性条件下给出了达到上界的收敛速率。我们还引入了 \(\beta \)-reach,这是对 reach 的概括,它排除了大小小于参数 \(\beta \in [0,\infty )\) 的小尺度特征。数值研究表明了如何在高维度上使用(beta)-reach来推断光滑子实体的reach和其他几何特性。
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