# Random polytopes and the wet part for arbitrary probability distributions

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
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.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.inria.fr/hal-02050632
Contributor : Xavier Goaoc <>
Submitted on : Wednesday, February 27, 2019 - 11:52:18 AM
Last modification on : Friday, June 7, 2019 - 4:32:47 PM

### Identifiers

• HAL Id : hal-02050632, version 1
• ARXIV : 1902.06519

### Citation

Imre Barany, Matthieu Fradelizi, Xavier Goaoc, Alfredo Hubard, Günter Rote. Random polytopes and the wet part for arbitrary probability distributions. 2019. ⟨hal-02050632⟩

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