  Let X_{d,n} be an n-element subset of {0,1}^d chosen uniformly at random, and
denote by P_{d,n} := conv X_{d,n} its convex hull. Let D_{d,n} be the density
of the graph of P_{d,n} (i.e., the number of one-dimensional faces of P_{d,n}
divided by n(n-1)/2). Our main result is that, for any function n(d), the
expected value of D_{d,n(d)} converges (with d tending to infinity) to one if,
for some arbitrary e > 0, n(d) <= (\sqrt{2}-e)^d holds for all large d, while
it converges to zero if n(d) >= (\sqrt{2}+e)^d holds for all large d.

Combinatorics;Optimization and Control;Probability