Several measures of non-convexity (departures from convexity) have been introduced in the literature, both for sets and functions. Some of them are of geometric nature, while others are more of topological nature. We address the statistical analysis of some of these measures of non-convexity of a set $S$, by dealing with their estimation based on a sample of points in $S$. We introduce also a new measure of non-convexity. We discuss briefly about these different notions of non-convexity, prove consistency and find the asymptotic distribution for the proposed estimators. We also consider the practical implementation of these estimators and illustrate their applicability to a real data example.

翻译：在文献中,对组合和功能都采用了若干非集中度的计量方法,其中有些是几何性质,而另一些则更具有地貌性质。我们处理对一些非集中度的计量方法的统计分析,方法是根据以美元为单位的点数抽样进行估算。我们还采用一种新的非集中度的计量方法。我们简要讨论这些不同的非集中度概念,证明一致性,并为拟议的估算者找到无症状的分布。我们还考虑这些估算方法的实际实施,并展示其对一个真实数据实例的适用性。