On the deterministic interior body of random polytopes

Abstract

Let be a sequence of independent copies of a random vector in . We revisit the question to determine the asymptotic shape of the random polytope where . We show that for any there exists a constant such that the following holds true: if is a Borel probability measure on then, for all we have that with probability greater than , where is the convex set of all points with half‐space depth greater than or equal to . Our approach does not require any additional assumptions about the measure and hence it generalizes and/or improves a sequence of previous results. Moreover, for the class of strongly regular measures we compare the family to other natural families of convex bodies associated with , such as the ‐centroid bodies of or the level sets of the Cramér transform of , and use this information in order to estimate the size of a random .