Abstract. 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 ∆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 ). Our main result is that, for any function 2
n(d), the expected value of ∆d,n(d) converges (with d → ∞) to one if, √
for some arbitrary ε < 0, n(d) ≤ ( 2 − ε)d holds for all large d, while it √
converges to zero if n(d) ≥ ( 2 + ε)d holds for all large d.