Journal of Theoretical Probability

, Volume 24, Issue 2, pp 330–341 | Cite as

The Mean Perimeter of Some Random Plane Convex Sets Generated by a Brownian Motion

  • Philippe Biane
  • Gérard Letac


If C 1 is the convex hull of the curve of a standard Brownian motion in the complex plane watched from 0 to 1, we consider the convex hulls of C 1 and several rotations of it and compute the mean of the length of their perimeter by elementary calculations. This can be seen geometrically as a study of the exit time by a Brownian motion from certain polytopes having the unit circle as an inscribed one.


Stopping times Exit times Random convex sets Brownian motion 

Mathematics Subject Classification (2000)

60J65 52A10 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.CNRS, Laboratoire d’Informatique, Institut Gaspard MongeUniversité Paris-EstMarne-la-Vallée Cedex 2France
  2. 2.Laboratoire de Statistique et ProbabilitésUniversité Paul SabatierToulouseFrance

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