Electronic Journal of Probability

Volumetric properties of the convex hull of an $n$-dimensional Brownian motion

Ronen Eldan

Full-text: Open access

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.

Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 45, 34 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065687

Digital Object Identifier
doi:10.1214/EJP.v19-2571

Mathematical Reviews number (MathSciNet)
MR3210546

Zentralblatt MATH identifier
1298.52005

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Eldan, Ronen. Volumetric properties of the convex hull of an $n$-dimensional Brownian motion. Electron. J. Probab. 19 (2014), paper no. 45, 34 pp. doi:10.1214/EJP.v19-2571. https://projecteuclid.org/euclid.ejp/1465065687


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