The reach of a set $M \subset \mathbb R^d$, also known as condition number when $M$ is a manifold, was introduced by Federer in 1959 and is a central concept in geometric measure theory, set estimation, manifold learning, among others areas. We introduce a universally consistent estimate of the reach, just assuming that the reach is positive. A necessary condition for the universal convergence of the reach is that the Haussdorf distance between the sample and the set converges to zero. Without further assumptions we show that the convergence rate of this distance can be arbitrarily slow. However, under a weak additional assumption, we provide rates of convergence for the reach estimator. We also show that it is not possible to determine if the reach of the support of a density is zero or not based on a finite sample. We provide a small simulation study and a bias correction method for the case when $M$ is a manifold.

翻译：设定 $M \ subset \ subthbb \ mathb R ⁇ d$ 的覆盖范围, 也称为条件号, 当 $M 是一元时, 由Federerer 于1959年推出, 是几何测量理论、 设定估算、 多重学习, 以及其它领域的中心概念 。 我们引入了普遍一致的估计范围, 只是假设该范围是正的 。 普及范围的一个必要条件是, 样本和集的距离会趋同为零 。 没有进一步的假设, 我们显示这一距离的趋同率可以任意地缓慢。 但是, 在一个薄弱的附加假设下, 我们为目标估计器提供了趋同率 。 我们还表明, 无法根据一定的样本确定密度支持的覆盖范围是否为零或不是零 。 我们为当 $ 数 时 的情况提供小型模拟研究和偏差校正方法 。