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

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1.	Title	Title of document	Volumetric properties of the convex hull of an n-dimensional Brownian motion
2.	Creator	Author's name, affiliation, country	Ronen Eldan; Microsoft Research; United States
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)
4.	Description	Abstract	Let K be the convex hull of the path of a standard brownian motion B(t) in R^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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	The Israel Science foundation
7.	Date	(YYYY-MM-DD)	2014-05-19
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/2571
10.	Identifier	Digital Object Identifier	10.1214/EJP.v19-2571
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 19
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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