Bounds on the size of the convex hull of planar Brownian motion and related inverse processes

TL;DR

This paper derives bounds on the expected geometric quantities of the convex hull of planar Brownian motion, including new bounds on circumradius and inradius, and investigates the growth rates of related inverse processes.

Contribution

It provides the first bounds on the expected inradius and circumradius of the Brownian convex hull and analyzes the growth of inverse processes related to its size.

Findings

Bounds on expected inradius and circumradius established.

Expected perimeter and area are known explicitly.

Bounds on inverse processes reveal growth rates of the convex hull.

Abstract

We establish bounds on expected values of various geometric quantities that describe the size of the convex hull spanned by a path of the standard planar Brownian motion. Expected values of the perimeter and the area of the Brownian convex hull are known explicitly, and satisfactory bounds on the expected value of its diameter can be found in the literature as well. In this work we investigate circumradius and inradius of the Brownian convex hull and obtain lower and upper bounds on their expected values. Our other goal is to find bounds on the related inverse processes (that correspond to the perimeter, area, diameter, circumradius and inradius of the convex hull) which provide us with some information on the speed of growth of the size of the Brownian convex hull.