@article{EJP2571,
author = {Ronen Eldan},
title = {Volumetric properties of the convex hull of an n-dimensional Brownian motion},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {19},
year = {2014},
keywords = {},
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.},
pages = {no. 45, 1-34},
issn = {1083-6489},
doi = {10.1214/EJP.v19-2571},
url = {http://ejp.ejpecp.org/article/view/2571}}