Abstract
We supposeK(w) to be the boundary of the closed convex hull of a sample path ofZ t(w), 0 ≦t ≦ 1 of Brownian motion ind-dimensions. A combinatorial result of Baxter and Borndorff Neilson on the convex hull of a random walk, and a limiting process utilizing results of P. Levy on the continuity properties ofZ t(w) are used to show that the curvature ofK(w) is concentrated on a metrically small set.
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References
O. Barndorff-Nielson and G. Baxter,Combinational lemmas in n-dimensions, Trans. Amer. Math. Soc.108 (1963), 313–325.
L. Carleson,On a class of meromorphic functions and its associated exceptional sets. Thesis, University of Uppsala, (1950).
J. R. Kinney,The convex hull of plane Brownian motion, Ann. Math. Statist.34 (1963), 327–329.
P. Lévy,Processus stochastiques et mouvement Brownien, Gautier-Villars, Paris, 1948.
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Research received support from ONR under contract No. Nonr-2587(02).
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Kinney, J.R. Convex hull of Brownian motion ind-dimensions. Israel J. Math. 4, 139–143 (1966). https://doi.org/10.1007/BF02937459
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DOI: https://doi.org/10.1007/BF02937459