Expected perimeter of the convex hull of planar Brownian motion stopped upon exiting the unit disk

TL;DR

This paper computes the exact expected perimeter of the convex hull formed by planar Brownian motion stopped upon exiting the unit disk, using harmonic measure and probabilistic techniques.

Contribution

It provides the first exact formula for the expected perimeter of the convex hull of planar Brownian motion in a bounded domain.

Findings

Derived an exact expression for the expected perimeter.

Obtained bounds and Monte Carlo estimates for the expected area.

Analyzed related hulls like star hull and topological hull.

Abstract

We study the convex hull of planar Brownian motion run until the exit time from the unit disk. Our primary objective is to compute the expected perimeter of this convex hull, thereby complementing recent results on the convex hull of reflecting Brownian motion in confined geometries. We reduce the problem to computing the expected value of the Brownian motion's maximum horizontal displacement at the exit time, and then recast this maximum in terms of harmonic measure in a domain we call the truncated disk. In particular, we obtain an exact expression for the expected perimeter. We also obtain nontrivial bounds and a Monte Carlo estimate for the expected area of this convex hull and comment on why computing the exact expected area is a much harder problem. We conclude with further results on the expected areas of two related hulls of the path run until exiting the disk, namely, the star hull and topological hull.