Probability Theory and Related Fields

, Volume 79, Issue 3, pp 327-368

First online:

Limit theorems for convex hulls

  • Piet GroeneboomAffiliated withFaculty of Mathematics and Informatics, Delft University of Technology

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It is shown that the process of vertices of the convex hull of a uniform sample from the interior of a convex polygon converges locally, after rescaling, to a strongly mixing Markov process, as the sample size tends to infinity. The structure of the limiting Markov process is determined explicitly, and from this a central limit theorem for the number of vertices of the convex hull is derived. Similar results are given for uniform samples from the unit disk.