Brownian limits, local limits and variance asymptotics for convex hulls in the ball

TL;DR

This paper investigates the asymptotic behavior of convex hulls in the unit ball, establishing limit theorems and variance asymptotics for geometric functionals using growth processes and stabilization theory.

Contribution

It introduces the paraboloid hull process and derives explicit asymptotic expressions and limit theorems for convex hull functionals in high-density Poisson samples.

Findings

Brownian sheet limits for defect volume and mean width

Explicit variance asymptotics for k-face and intrinsic volume functionals

Local limit theorems for scaled radius-vector and support functions

Abstract

Schreiber and Yukich [Ann. Probab. 36 (2008) 363-396] establish an asymptotic representation for random convex polytope geometry in the unit ball $\mathbb{B}^d, d\geq2$, in terms of the general theory of stabilizing functionals of Poisson point processes as well as in terms of generalized paraboloid growth processes. This paper further exploits this connection, introducing also a dual object termed the paraboloid hull process. Via these growth processes we establish local functional limit theorems for the properly scaled radius-vector and support functions of convex polytopes generated by high-density Poisson samples. We show that direct methods lead to explicit asymptotic expressions for the fidis of the properly scaled radius-vector and support functions. Generalized paraboloid growth processes, coupled with general techniques of stabilization theory, yield Brownian sheet limits for the defect volume and mean width functionals. Finally we provide explicit variance asymptotics and central limit theorems for the k-face and intrinsic volume functionals.