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Random polytopes and the wet part for arbitrary probability distributions

Imre Barany 1 Matthieu Fradelizi 2 Xavier Goaoc 3 Alfredo Hubard 4 Günter Rote 5
1 Alfréd Rényi Institute of Mathematics
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 GAMBLE - Geometric Algorithms and Models Beyond the Linear and Euclidean realm
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
4 LIGM - Laboratoire d'Informatique Gaspard-Monge
5 Institut für Informatik [Berlin]
Abstract : We examine how the measure and the number of vertices of the convex hull of a random sample of $n$ points from an arbitrary probability measure in $\mathbf{R}^d$ relates to the wet part of that measure. This extends classical results for the uniform distribution from a convex set [B\'ar\'any and Larman 1988]. The lower bound of B\'ar\'any and Larman continues to hold in the general setting, but the upper bound must be relaxed by a factor of $\log n$. We show by an example that this is tight.
Keywords : Epsilon-nets Floating body Random polytopes
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https://hal.inria.fr/hal-02937527
Contributor : Xavier Goaoc <>
Submitted on : Monday, September 14, 2020 - 10:11:25 AM
Last modification on : Monday, March 22, 2021 - 2:27:06 PM

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Imre Barany, Matthieu Fradelizi, Xavier Goaoc, Alfredo Hubard, Günter Rote. Random polytopes and the wet part for arbitrary probability distributions. Annales Henri Lebesgue, UFR de Mathématiques - IRMAR, 2020, 3, pp.701-715. ⟨10.5802/ahl.44⟩. ⟨hal-02937527⟩

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