  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.
