@article{ECP2469,
author = {Olivier Devillers and Marc Glisse and Xavier Goaoc and Guillaume Moroz and Matthias Reitzner},
title = {The monotonicity of f-vectors of random polytopes},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {18},
year = {2013},
keywords = {Computational geometry;Convex hull;Complexity},
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.},
pages = {no. 23, 1-8},
issn = {1083-589X},
doi = {10.1214/ECP.v18-2469},
url = {http://ecp.ejpecp.org/article/view/2469}}