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

## TL;DR

This paper provides exact formulas and asymptotic approximations for the expected volumes of Gaussian polytopes, their intrinsic volumes, and external angles, connecting these to classical polytopes and order statistics of Gaussian variables.

## Contribution

It introduces precise formulas for Gaussian polytope volumes, relates intrinsic volumes to Gaussian order statistics, and extends results to heteroscedastic and deterministic polytopes.

## Key findings

- Exact expected volumes of Gaussian polytopes and zonotopes derived
- Asymptotic formulas improve previous estimates
- Intrinsic volumes linked to Gaussian maximum order statistics

## Abstract

Let $X_1,\ldots,X_n$ be a standard normal sample in $\mathbb R^d$. We compute exactly the expected volume of the Gaussian polytope $\mathrm{conv}[X_1,\ldots,X_n]$, the symmetric Gaussian polytope $\mathrm{conv}[\pm X_1,\ldots,\pm X_n]$, and the Gaussian zonotope $[0,X_1]+\ldots+[0,X_n]$ by exploiting their connection to the regular simplex, the regular crosspolytope, 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 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 $ \mathrm{conv}[l_1X_1,\ldots,l_nX_n] $ and $ \mathrm{conv}[\pm l_1 X_1,\ldots, \pm l_n X_n], $ where $l_1,\ldots,l_n\geq 0$ are parameters, and the intrinsic volumes of the corresponding deterministic polytopes. Finally, we relate the $k$-th intrinsic volume of the regular simplex $S^{n-1}$ to the expected maximum of independent standard Gaussian random variables $\xi_1,\ldots,\xi_n$ given that the maximum has multiplicity $k$. Namely, we show that $$ V_k(S^{n-1}) = \frac {(2\pi)^{\frac k2}} {k!} \cdot \lim_{\varepsilon\downarrow 0} \varepsilon^{1-k} \mathbb E [\max\{\xi_1,\ldots,\xi_n\} 1_{\{\xi_{(n)} - \xi_{(n-k+1)}\leq \varepsilon\}}], $$ where $\xi_{(1)} \leq \ldots \leq \xi_{(n)}$ denote the order statistics. A similar result holds for the crosspolytope if we replace $\xi_1,\ldots,\xi_n$ by their absolute values.