We examine how the measure and the number of vertices of the convex hull of a random sample of points from an arbitrary probability measure in relate to the wet part of that measure. This extends classical results for the uniform distribution from a convex set proved by Bárány and Larman in 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 . We show by an example that this is tight.
[Bee15] Random polytopes (2015) (https://repositorium.ub.uni-osnabrueck.de/bitstream/urn:nbn:de:gbv:700-2015062313276/1/thesis_beermann.pdf) (Ph. D. Thesis)
[BR17] Monotonicity of functionals of random polytopes (2017) (https://arxiv.org/abs/1706.08342)
[KTZ19] Beta polytopes and Poisson polyhedra: -vectors and angles (2019) (https://arxiv.org/abs/1805.01338)
[PA95] Combinatorial Geometry, John Wiley & Sons, 1995 | Zbl 0881.52001