Expected volumes of Gaussian polytopes, external angles, and multiple order statistics

Abstract

Let X 1 X_1 , …, X n X_n be a standard normal sample in R d \mathbb {R}^d . We compute exactly the expected volume of the Gaussian polytope conv ⁡ [ X 1 , … , X n ] \operatorname {conv} [X_1,\ldots ,X_n] , the symmetric Gaussian polytope conv ⁡ [ ± X 1 , … , ± X n ] \operatorname {conv} [\pm X_1,\ldots ,\pm X_n] , and the Gaussian zonotope [ 0 , X 1 ] + ⋯ + [ 0 , X n ] [0,X_1]+\cdots +[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 conv ⁡ [ l 1 X 1 , … , l n X n ] \operatorname {conv}[l_1X_1,\ldots ,l_nX_n] and conv ⁡ [ ± l 1 X 1 , … , ± l n X n ] \operatorname {conv} [\pm l_1 X_1,\ldots , \pm l_n X_n] , where l 1 l_1 , …, l n ≥ 0 l_n\geq 0 are parameters, and the intrinsic volumes of the corresponding deterministic polytopes. Finally, we relate the k k th intrinsic volume of the regular simplex Δ n − 1 \Delta ^{n-1} to the expected maximum of independent standard Gaussian random variables ξ 1 \xi _1 , …, ξ n \xi _n given that the maximum has multiplicity k k . Namely, we show that \[ V k ( Δ n − 1 ) = ( 2 π ) k 2 k ! ⋅ lim ε ↓ 0 ε 1 − k E [ max { ξ 1 , … , ξ n } 1 { ξ ( n ) − ξ ( n − k + 1 ) ≤ ε } ] , V_k(\Delta ^{n-1}) = \frac {(2\pi )^{\frac k2}} {k!} \cdot \lim _{\varepsilon \downarrow 0} \varepsilon ^{1-k} \mathbb {E} [\max \{\xi _1,\ldots ,\xi _n\} \mathbb {1}_{\{\xi _{(n)} - \xi _{(n-k+1)}\leq \varepsilon \}}], \] where ξ ( 1 ) ≤ ⋯ ≤ ξ ( n ) \xi _{(1)} \leq \cdots \leq \xi _{(n)} denote the order statistics. A similar result holds for the cross-polytope if we replace ξ 1 \xi _1 , …, ξ n \xi _n with their absolute values.