Peelings and Wrappings of Families of Convex Sets with Applications to Strongly Convex Sets Generated by Random Samples

Abstract

We introduce and study peeling and wrapping operations for families of compact convex sets. The two peeling procedures considered in the paper are the $m$-point peeling, obtained by intersecting the convex hulls remaining after all possible deletions of $m$ members of the family, and the recursive convex hull peeling, obtained by repeatedly removing the contributing sets, that is, those members whose deletion strictly changes the convex hull. Using polarity, we also introduce the dual wrapping operations for intersections of convex sets. The deterministic part of the paper develops the geometric framework needed for these constructions. In particular, we study contributing sets under general position assumptions, explain the role of compactness of convex hulls of subfamilies, and prove continuity results for both peeling procedures with respect to a suitable vague convergence of locally finite point measures on the space of compact convex sets. The probabilistic part applies this framework to $K$-hulls generated by random samples from a convex body $K$. Assuming that $K$ is strictly convex and regular, we prove that the m-point and recursive peelings of the polar bodies associated with the random $K$-hulls converge in distribution to the corresponding peelings of the limiting Poisson object. By polarity, this also yields distributional convergence of the associated wrapping operations for the rescaled random sets themselves.