Convex hulls of perturbed random point sets

TL;DR

This paper analyzes the convex hulls of perturbed random points, revealing phase transitions in their geometric properties and establishing limit theorems for face counts under various perturbation regimes.

Contribution

It introduces a new model of perturbed random points, characterizes the scaling limits, and derives explicit asymptotics and limit theorems for convex hull features.

Findings

Convergence to Poisson point processes in five regimes based on perturbation parameter.

Explicit asymptotics for the expected number of k-faces of the convex hull.

Variance asymptotics and CLT for the number of k-faces in Poisson case.

Abstract

We consider the convex hull of the perturbed point process comprised of $n$ i.i.d. points, each distributed as the sum of a uniform point on the unit sphere $\S^{d-1}$ and a uniform point in the $d$-dimensional ball centered at the origin and of radius $n^{\alpha}, \alpha \in (-\infty, \infty)$. This model, inspired by the smoothed complexity analysis introduced in computational geometry \cite{DGGT,ST}, is a perturbation of the classical random polytope. We show that the perturbed point process, after rescaling, converges in the scaling limit to one of five Poisson point processes according to whether $\alpha$ belongs to one of five regimes. The intensity measure of the limit Poisson point process undergoes a transition at the values $\alpha = \frac{-2} {d -1}$ and $\alpha = \frac{2} {d + 1}$ and it gives rise to four rescalings for the $k$-face functional on perturbed data. These rescalings are used to establish explicit expectation asymptotics for the number of $k$-dimensional faces of the convex hull of either perturbed binomial or Poisson data. In the case of Poisson input, we establish explicit variance asymptotics and a central limit theorem for the number of $k$-dimensional faces. Finally it is shown that the rescaled boundary of the convex hull of the perturbed point process converges to the boundary of a parabolic hull process.