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Bounds on some geometric functionals of high dimensional Brownian convex hulls and their inverse processes (2407.08712v3)
Published 11 Jul 2024 in math.PR and math.MG
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 $n$.