EXPECTED VOLUMES OF GAUSSIAN POLYTOPES, EXTERNAL ANGLES, AND MULTIPLE ORDER STATISTICS

Let X-1, ..., X-n be a standard normal sample in R-d. We compute exactly the expected volume of the Gaussian polytope cony, [X-1, ..., X-n], the symmetric Gaussian polytope cony [+/- X-1, ..., +/- X-n] and the Gaussian zonotope [0, X-1] + ... + [0, X-n] by exploiting their connection to the regular simplex, the regular cross-polytope, and the cube with the aid of Tsirelson's formula. The expected volumes of these random polytopes are given by essentially the same expressions as the intrinsic volumes and external angles of the regular polytopes. For all of these quantities, we obtain asymptotic formulae which are more precise than the results which were known before. More generally, we determine the expected volumes of some heteroscedastic random polytopes including cony [l(1)X(1), ..., l(n)X(n)] and cony [+/- l(1)X(1), ..., +/- l(n)X(n)], where l(1), ..., l(n) >= 0 are parameters, and the intrinsic volumes of the corresponding deterministic polytopes. Finally, we relate the kth intrinsic volume of the regular simplex Delta(n-1) to the expected maximum of independent standard Gaussian random variables xi(1), ....,xi(n) n given that the maximum has multiplicity k. Namely, we show that V-k(Delta(n-1)) = (2 pi)k/2/k! lim epsilon down arrow 0 epsilon(1-k) E vertical bar max {xi(1),... xi(n)} xi((n)) - xi(n-k+1) <= epsilon}]. where xi((1)) <= ... <= xi((n)) denote the order statistics. A similar result holds for the cross-polytope if we xi(1), ...., xi(n) replace with their absolute values.