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

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), Jovalekić (2021), and Cygan, Šebek, 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 $n$.

💡 Research Summary

This paper investigates the geometric size of the convex hull generated by a standard n‑dimensional Brownian motion started at the origin. For the convex hull (H_t=\operatorname{conv}{W_s:0\le s\le t}) the authors consider four basic functionals: volume (V_t), surface area (S_t), diameter (D_t) and circumradius (R_t). Because each functional is almost surely non‑decreasing in time, the associated right‑continuous inverse processes are defined by
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