Facial structure of strongly convex sets generated by random samples

• Published in: Advances in mathematics (New York. 1965) 2022-02, Vol.395, p.108086, Article 108086
• Main Authors: Marynych, Alexander, Molchanov, Ilya
• Format: Article
• Language: English
• Subjects: Ball convexity; f-vector; Facial structure; K-strong convexity; Random convex bodies; Zero cell of a Poisson tessellation
• ISSN: 0001-8708; 1090-2082
• DOI: 10.1016/j.aim.2021.108086

The K-hull of a compact set A⊆Rd, where K⊆Rd is a fixed compact convex body, is the intersection of all translates of K that contain A. A set is called K-strongly convex if it coincides with its K-hull. We propose a general approach to the analysis of facial structure of K-strongly convex sets, similar to the well developed theory for polytopes, by introducing the notion of k-dimensional faces, for all k=0,…,d−1. We then apply our theory in the case when A=Ξn is a sample of n points picked uniformly at random from K. We show that in this case the set of x∈Rd such that x+K contains the sample Ξn, upon multiplying by n, converges in distribution to the zero cell of a certain Poisson hyperplane tessellation. From these results we deduce convergence in distribution of the corresponding f-vector of the K-hull of Ξn to a certain limiting random vector, without any normalisation, and also the convergence of all moments of the f-vector.