Limit theory for the first layers of the random convex hull peeling in a simple polytope

TL;DR

This paper investigates the asymptotic behavior of the first layers of convex hull peeling inside a simple polytope, providing limit theorems for key geometric functionals as the point process intensity increases.

Contribution

It extends limit theory from convex hulls to multiple peeling layers within simple polytopes, revealing layer-independent growth rates and establishing new asymptotic results.

Findings

Asymptotic limits for expected number of faces and volume defect

Central limit theorem for layer functionals

Growth rates are layer-independent in simple polytopes

Abstract

The convex hull peeling of a point set consists in taking the convex hull, then removing the extreme points and iterating that procedure until no point remains. The boundary of each hull is called a layer. Following on from [15], we study the first layers generated by the peeling procedure when the point set is chosen as a homogeneous Poisson point process inside a polytope when the intensity goes to infinity. We focus on some specific functionals, namely the number of k-dimensional faces and the outer defect volume. Since the early works of R{\'e}nyi and Sulanke, it is well known that both the techniques and the rates are completely different for the convex hull when the underlying convex body has a smooth boundary or when it is itself a polytope. We expect such dichotomy to extend to the further layers of the peeling. More precisely we provide asymptotic limits for their expectation and variance as well as a central limit theorem. In particular, as in the unit ball, the growth rates do not depend on the layer. The method builds upon previous constructions for the convex hull contained in [9] and [18] and requires the assumption that the underlying polytope is simple.