On the graph-density of random 0/1-polytopes [article]

Volker Kaibel, Anja Remshagen
2003 arXiv   pre-print
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) <= (√(2)-e)^d holds for all large d, while it converges to zero if n(d) >= (√(2)+e)^d holds for all large d.
arXiv:math/0306246v1 fatcat:tlqtdrkkkvgshdysrfcpmwgkf4