论文标题
$ r $ -CONVEX套件，不符合本地签约
An $r$-convex set which is not locally contractible
论文作者
论文摘要
$ \ mathbb {r}^d $子集的形状限制的研究在许多领域都有多个应用程序，具有凸度，$ r $ - 概念和正面覆盖范围，其中一些最著名的，通常在集合估算中强加。 J. Perkal在1956年归因于K. Borsuk的以下问题：查找不可本地签约的$ R $ -CONVEX套件。以这种方式说明找到这样的集合是微不足道的。但是，如果我们要求该集合等于其内部的关闭（例如，如果该集合是对概率分布的支持，则相对于$ d $ d $维度的Lebesgue措施绝对连续），那么问题要困难得多。我们提供了一个不易于签约的集合的反示例，即$ r $ -Convex。这也证明，绝对连续分布的积极覆盖范围的支持类别包括$ r $ -Convex支持的类别。
The study of shape restrictions of subsets of $\mathbb{R}^d$ have several applications in many areas, being convexity, $r$-convexity, and positive reach, some of the most famous, and typically imposed in set estimation. The following problem was attributed to K. Borsuk, by J. Perkal in 1956: find an $r$-convex set which is not locally contractible. Stated in that way is trivial to find such a set. However, if we ask the set to be equal to the closure of its interior (a condition fulfilled for instance if the set is the support of a probability distribution absolutely continuous with respect to the $d$-dimensional Lebesgue measure), the problem is much more difficult. We present a counter example of a not-locally contractible set, which is $r$-convex. This also proves that the class of supports with positive reach of absolutely continuous distributions includes strictly the class of $r$-convex supports.
