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2014 Volumetric properties of the convex hull of an $n$-dimensional Brownian motion
Ronen Eldan
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Electron. J. Probab. 19: 1-34 (2014). DOI: 10.1214/EJP.v19-2571

Abstract

Let $K$ be the convex hull of the path of a standard brownian motion $B(t)$ in $\mathbbR^n$, taken at time $0 < t < 1$. We derive formulas for the expected volume and surface area of $K$. Moreover, we show that in order to approximate $K$ by a discrete version of $K$, namely by the convex hull of a random walk attained by taking $B(t_n)$ at discrete (random) times, the number of steps that one should take in order for the volume of the difference to be relatively small is of order $n^3$. Next, we show that the distribution of facets of $K$ is in some sense scale invariant: for any given family of simplices (satisfying some compactness condition), one expects to find in this family a constant number of facets of $tK$ as $t$ approaches infinity. Finally, we discuss some possible extensions of our methods and suggest some further research.

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Ronen Eldan. "Volumetric properties of the convex hull of an $n$-dimensional Brownian motion." Electron. J. Probab. 19 1 - 34, 2014. https://doi.org/10.1214/EJP.v19-2571

Information

Accepted: 19 May 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1298.52005
MathSciNet: MR3210546
Digital Object Identifier: 10.1214/EJP.v19-2571

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