The mean perimeter of some random plane convex sets generated by a Brownian motion

TL;DR

This paper calculates the average perimeter length of convex hulls formed by Brownian motion paths and their rotations in the complex plane, providing explicit elementary calculations for these geometric properties.

Contribution

It introduces a method to compute the mean perimeter of convex hulls generated by Brownian motion and its rotations, offering explicit elementary formulas.

Findings

Explicit formulas for the mean perimeter of convex hulls of Brownian motion paths.

Elementary calculations for geometric properties of Brownian convex hulls.

Analysis of rotations of Brownian convex hulls and their perimeter statistics.

Abstract

If $C_1$ is the convex hull of the curve of the standard Brownian motion in the complex plane watched from 0 to 1, we consider the convex hulls of $C_1$ and several rotations of it and we compute the mean of the length of their perimeter by elementary calculations.