VOLUMES SPANNED BY RANDOM POINTS IN THE HYPERCUBE

Consider the hypercube [0, 1]n in R(n). This has 2n vertices and volume 1. Pick N = N(n) vertices independently at random, form their convex hull, and let V(n) be its expected volume. How large should N(n) be to pick up significant volume? Let kappa = 2/square-root e almost-equal-to 1.213, and let epsilon > 0. We shall show that, as n --> infinity, V(n) --> if N(n) less-than-or-equal-to (kappa - epsilon)n, and V(n) --> 1 if N(n) greater-than-or-equal-to (kappa + epsilon)n. A similar result holds for sampling uniformly from within the hypercube, with constant lambda = exp{integral-0/infinity(1/u - 1/(e(u) - 1))2 du} almost-equal-to 2.136.