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

Let Xd,n be an n-element subset of {0, 1} chosen uniformly at random, and denote by Pd,n := conv Xd,n its convex hull. Let ∆d,n be the density of the graph of Pd,n (i.e., the number of one-dimensional faces of Pd,n divided by ` n 2 ́ ). Our main result is that, for any function n(d), the expected value of ∆d,n(d) converges (with d → ∞) to one if, for some arbitrary ε > 0, n(d) ≤ ( √ 2 − ε) holds for all large d, while it converges to zero if n(d) ≥ ( √ 2 + ε) holds for all large d.