Bounds on some geometric functionals of high dimensional Brownian convex hulls and their inverse processes

TL;DR

This paper establishes bounds on geometric functionals of high-dimensional Brownian convex hulls and their inverse processes, extending previous planar results to higher dimensions with precise asymptotic behavior.

Contribution

It introduces a novel embedding procedure for an n-simplex within the convex hull of Brownian motion, providing sharp bounds in high dimensions.

Findings

Expected time for convex hull to reach unit volume is at most n times the n-th root of n!

Bounds accurately reflect asymptotic growth or decay in dimension n

Extends previous planar results to higher-dimensional settings

Abstract

We prove two-sided bounds on the expected values of several geometric functionals of the convex hull of Brownian motion in $\mathbb{R}^n$ and their inverse processes. This extends some recent results of McRedmond and Xu (2017), Jovaleki\'{c} (2021), and Cygan, \v{S}ebek, and the first author (2023) from the plane to higher dimensions. Our main result shows that the average time required for the convex hull in $\mathbb{R}^n$ to attain unit volume is at most $n\sqrt[n]{n!}$. The proof relies on a novel procedure that embeds an $n$-simplex of prescribed volume within the convex hull of the Brownian path run up to a certain stopping time. All of our bounds capture the correct order of asymptotic growth or decay in the dimension $n$.