Facial structure of strongly convex sets generated by random samples

TL;DR

This paper develops a theory for the facial structure of strongly convex sets generated by random samples, showing convergence of geometric features to a Poisson hyperplane tessellation in high probability.

Contribution

It introduces a new framework for analyzing the facial structure of $K$-strongly convex sets and applies it to random samples, establishing convergence results without normalization.

Findings

Convergence of the set of points defining the $K$-hull to a Poisson hyperplane tessellation.

Distributional convergence of the $f$-vector of the $K$-hull.

All moments of the $f$-vector also converge.

Abstract

The $K$-hull of a compact set $A\subset\mathbb{R}^d$, where $K\subset \mathbb{R}^d$ 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 $k=0,\dots,d-1$. We then apply our theory in the case when $A=\Xi_n$ is a sample of $n$ points picked uniformly at random from $K$. We show that in this case the set of $x\in\mathbb{R}^d$ such that $x+K$ contains the sample $\Xi_n$, upon multiplying by $n$, converges in distribution to the zero cell of a certain Poisson hyperplane tessellation. From this results we deduce convergence in distribution of the corresponding $f$-vector of the $K$-hull of $\Xi_n$ to a certain limiting random vector, without any normalisation, and also the convergence of all moments of the $f$-vector.