Skip to Main content Skip to Navigation
Journal articles

The monotonicity of $f$-vectors of random polytopes

Olivier Devillers 1 Marc Glisse 1 Xavier Goaoc 2 Guillaume Moroz 2 Matthias Reitzner 3
1 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
2 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry, Inria Nancy - Grand Est
3 Institut für Mathematik
FB6/Institut für Mathematik - Institut für Mathematik [Osnabrück]
Abstract : Let K be a compact convex body in $Rd$, let $Kn$ be the convex hull of n points chosen uniformly and independently in K, and let $fi(Kn)$ denote the number of i-dimensional faces of $Kn$. We show that for planar convex sets, $E[f0(Kn)]$ is increasing in $n$. In dimension $d≥3 we prove that if limn→∞ E[fd−1(Kn)]Anc=1$ for some constants A and c>0 then the function $n↦E[fd−1(Kn)]$ is increasing for n large enough. In particular, the number of facets of the convex hull of n random points distributed uniformly and independently in a smooth compact convex body is asymptotically increasing. Our proof relies on a random sampling argument.
Document type :
Journal articles
Complete list of metadata

Cited literature [16 references]  Display  Hide  Download
Contributor : Olivier Devillers Connect in order to contact the contributor
Submitted on : Thursday, November 24, 2016 - 3:40:44 PM
Last modification on : Friday, October 22, 2021 - 4:41:40 AM
Long-term archiving on: : Monday, March 20, 2017 - 9:01:47 PM


Publisher files allowed on an open archive




Olivier Devillers, Marc Glisse, Xavier Goaoc, Guillaume Moroz, Matthias Reitzner. The monotonicity of $f$-vectors of random polytopes. Electronic Communications in Probability, Institute of Mathematical Statistics (IMS), 2013, 18 (23), pp.1-8. ⟨10.1214/ECP.v18-2469⟩. ⟨hal-00805690⟩



Record views


Files downloads