Corollary 2

— Let E be a real vector space of finite dimension n, $(e_1, e_2, \ldots, e_n)$ a basis of E, P the set of $x = \sum_{i=1}^n \xi_i e_i \in E$ such that $0 \leq \xi_i \leq 1$ for all i, and $\mu$ the unique Haar measure on the additive group E such that $\mu(P) = 1$. Let $x_1, \ldots, x_n$ be points of E, S the closed convex envelope in E of the set $\{0, x_1, \ldots, x_n\}$. Writing $x_i = \sum_{j=1}^n \alpha_{ij} e_j$, one has

We shall identify E with $\mathbf{R}^n$ by means of the isomorphism that transforms $(e_i)$ into the canonical basis of $\mathbf{R}^n$. Then $\mu$ is identified with the Lebesgue measure $\mu_n$ on $\mathbf{R}^n$.

Suppose first that $x_i = e_i$ for all i. Then S is the set $S_n$ of $x = (\xi_i)$ in $\mathbf{R}^n$ such that

Set $\mu_n(S_n) = a_n$. Let $\lambda \in \mathbf{R}$. Identifying $\mathbf{R}^n$ with $\mathbf{R}^{n-1} \times \mathbf{R}$, we may consider the section $C_\lambda$ of $S_n$ at $\lambda$. This section is empty if $\lambda < 0$ or $\lambda > 1$; if $0 \leq \lambda \leq 1$, $C_\lambda$ is the set of $(\xi_1, \ldots, \xi_{n-1}) \in \mathbf{R}^{n-1}$ such that

hence may be deduced from $S_{n-1}$ by a homothety of ratio $1 - \lambda$, so that $\mu_{n-1}(C_\lambda) = (1 - \lambda)^{n-1} a_{n-1}$. By the Lebesgue–Fubini theorem,

Since $a_1 = 1$, one sees that $a_n = \frac{1}{n!}$.

Let us return to the general case of the corollary. Let $u$ be the endomorphism of $\mathbf{R}^n$ such that $u(e_i) = x_i$ for all $i$. One has $u(S_n) = S$. If $u$ is invertible, Prop. 15 proves that

Since $S - \overset{\circ}{S}$ is contained in a finite number of hyperplanes, $\mu(\overset{\circ}{S}) = \mu(S)$. Finally, if $u$ is not invertible, then $S$ is contained in a hyperplane, so that $\mu(S) = 0 = \det(\alpha_{ij})$.