2024/07/11 by Hugo Panzo, Panzo, Hugo, Evan Socher +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #60D05 #60J65 (Primary) 52A20 #90C25 (Secondary) #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2407.08712
openalex publication_date 2024/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove two-sided bounds on the expected values of several geometric functionals of the convex hull of Brownian motion in ℝn 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 ℝn to attain unit volume is at most n√[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.