On the Graph-Density of Random 0/1-Polytopes

Let Xd,n be an n-element subset of {0,1}d chosen uniformly at random, and denote by Pd,n: = conv Xd,n its convex hull. Let Δd,n be the density of the graph of Pd,n (i.e., the number of one-dimensional faces of Pd,n divided by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\binom{n}{2}$\end{document}). Our main result is that, for any function n(d), the expected value of Δd,n(d) converges (with d→ ∞) to one if, for some arbitrary ε> 0, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n(d)\le (\sqrt{2}-\varepsilon)^d$\end{document} holds for all large d, while it converges to zero if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n(d)\ge (\sqrt{2}+\varepsilon)^d$\end{document} holds for all large d.