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The monotonicity of f-vectors of random polytopes

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1.	Title	Title of document	The monotonicity of f-vectors of random polytopes
2.	Creator	Author's name, affiliation, country	Olivier Devillers; INRIA; France
2.	Creator	Author's name, affiliation, country	Marc Glisse; INRIA; France
2.	Creator	Author's name, affiliation, country	Xavier Goaoc; INRIA; France
2.	Creator	Author's name, affiliation, country	Guillaume Moroz; INRIA; France
2.	Creator	Author's name, affiliation, country	Matthias Reitzner; Universität Osnabrück.; Germany
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Computational geometry;Convex hull;Complexity
3.	Subject	Subject classification	60D05;52A22
4.	Description	Abstract	Let $K$ be a compact convex body in ${\mathbb R}^d$, let $K_n$ be the convex hull of $n$ points chosen uniformly and independently in $K$, and let $f_{i}(K_n)$ denote the number of $i$-dimensional faces of $K_n$. We show that for planar convex sets, $E[f_0 (K_n)]$ is increasing in $n$.  In dimension $d \geq 3$ we prove that if $\lim_{n \to \infty} \frac{E[f_{d-1}(K_n)]}{An^c}=1$ for some constants $A$ and $c>0$ then the function $n \mapsto E[f_{d-1}(K_n)]$ 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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)	ANR (french national agency for research)
7.	Date	(YYYY-MM-DD)	2013-03-28
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ecp.ejpecp.org/article/view/2469
10.	Identifier	Digital Object Identifier	10.1214/ECP.v18-2469
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 18
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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