The reach of a submanifold is a crucial regularity parameter for manifold learning and geometric inference from point clouds. This paper relates the reach of a submanifold to its convexity defect function. Using the stability properties of convexity defect functions, along with some new bounds and the recent submanifold estimator of Aamari and Levrard (Ann. Statist. 47(1), 177–204 (2019)), an estimator for the reach is given. A uniform expected loss bound over a \({\mathscr {C}}^k\) model is found. Lower bounds for the minimax rate for estimating the reach over these models are also provided. The estimator almost achieves these rates in the \({\mathscr {C}}^3\) and \({\mathscr {C}}^4\) cases, with a gap given by a logarithmic factor.

子流形的范围是流形学习和点云几何推断的关键规律性参数。本文将子流形的范围与其凸缺陷函数联系起来。使用凸缺陷函数的稳定性属性，以及一些新的界限和最近的 Aamari 和 Levrard 的子流形估计器 (Ann. Statist. 47 (1), 177–204 (2019))，给出了范围的估计器。找到了\({\mathscr {C}}^k\)模型上的统一预期损失。还提供了用于估计这些模型的范围的最小最大速率的下限。估计器在\({\mathscr {C}}^3\)和\({\mathscr {C}}^4\)情况下几乎达到了这些比率，并由对数因子给出差距。