Proposition 8

— For any family $(A_i)_{i \in I}$ of convex subsets of an affine space E, the convex envelope of $\bigcup_{i \in I} A_i$ is precisely the set of linear combinations $\sum_{i \in I} \lambda_i x_i$, where $x_i \in A_i, \lambda_i \geq 0$ for all $i \in I$ ($\lambda_i = 0$ except for finitely many indices) and $\sum_{i \in I} \lambda_i = 1$.

Denote the set of these linear combinations by C, clearly C is contained in every convex set which contains all the $A_i$ (II, p. 8, prop. 1); on the other hand $A_i \subset C$ for every $i$. All that remains to be proved is that C is convex. Let $x = \sum \lambda_i x_i$, $y = \sum \mu_i y_i$ be two points of C and $\alpha$ be a number such that $0 < \alpha < 1$, write γ_i = αλ_i + (1 - α) μ_i for every i ∈ I, and let J be the set (finite) of the indices of I for which γ_i ≠ 0. We can write αx + (1 - α)y = ∑_{i∈J} γ_i z_i, where

belongs to A_i for all i ∈ J; but $\sum_{i\in J} \gamma_i = \alpha \sum_{i\in I} \lambda_i + (1 - \alpha) \sum_{i\in I} \mu_i = 1$, and we see that $ax + (1 - \alpha) y \in C$. The proposition is proved.