Convex hulls of perturbed random point sets

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 ${\mathbb{S}}^{\mathit{d}-1}$ and a uniform point in the d-dimensional ball centered at the origin and of radius ${\mathit{n}}^{\mathit{\alpha }},\mathit{\alpha }\in (-\infty ,\infty )$. This model, inspired by the smoothed complexity analysis introduced in computational geometry (J. Comput. Geom. 7 (2016) 101–144; J. ACM 51 (2004) 385–463), 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 α belongs to one of five regimes. The intensity measure of the limit Poisson point process undergoes a transition at the values $\mathit{\alpha }=\frac{-2}{\mathit{d}-1}$ and $\mathit{\alpha }=\frac{2}{\mathit{d}+1}$ 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.