Limit theory for the first layers of the random convex hull peeling in the unit ball

TL;DR

This paper develops limit theorems for the geometric features of the initial layers in the convex hull peeling process of a Poisson point set in the unit ball, revealing layer-independent growth rates.

Contribution

It provides asymptotic expectations, variances, and a central limit theorem for the faces and volumes of the first peeling layers, a novel theoretical insight.

Findings

Asymptotic limits for expectations and variances of geometric features.

Layer-independent growth rates of the convex hull peeling.

Central limit theorem for the first layers' geometric quantities.

Abstract

The convex hull peeling of a point set is obtained by taking the convex hull of the set and repeating iteratively the operation on the interior points until no point remains. The boundary of each hull is called a layer. We study the number of k-dimensional faces and the outer defect intrinsic volumes of the first layers of the convex hull peeling of a homogeneous Poisson point process in the unit ball whose intensity goes to infinity. More precisely we provide asymptotic limits for their expectation and variance as well as a central limit theorem. In particular, the growth rates do not depend on the layer.