Imre Bárány, Matthieu Fradelizi, Xavier Goaoc, Alfredo Hubard, and Günter Rote:

Random polytopes and the wet part for arbitrary probability distributions

Annales Henri Lebesgue 3 (2020), 701–715. doi:10.5802/ahl.44, arXiv:1902.06519 [math.PR].  →BibTeX

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 d-dimensional space relates to the wet part of that measure. This extends classical results for the uniform distribution from a convex set [Bárány and Larman 1988]. The lower bound of Bárány 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.

The preprint on my homepage contains as an additional appendix a self-contained proof of the ε-net lemma from Komlós, Pach, and Woeginger [1992], which is used to show the existence of ε-nets.

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Last update: October 12, 2020.