Volumes Spanned by Random Points in the Hypercube

Consider the hypercube 0, 1]ⁿ in Rⁿ. 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 k=2/√≈1.213, and let ? > 0. We shall show that, as n→∞, V_n→0 if N(n)?(k??)ⁿ →1 if N(n) ? (k + ?)ⁿ. A similar result holds for sampling uniformly from within the hypercube, with constant .