Proposition 15

— Let $A_i$ ($1 \leq i \leq n$) be a finite number of compact convex sets in a Hausdorff topological vector space E. Then the convex envelope of the union of the $A_i$ is compact (and is, therefore, the same as the convex closed envelope of this union).

Let B be the compact set in $\mathbf{R}^n$ defined by the points $(\lambda_1, \lambda_2, ..., \lambda_n)$ where $\lambda_i \geq 0$ ($1 \leq i \leq n$), and $\sum_{i=1}^n \lambda_i = 1$. Define a continuous mapping of $B \times \prod_{i=1}^n A_i \subset \mathbf{R}^n \times E^n$ in E by the formula $(\lambda_1, \lambda_2, ..., \lambda_n, x_1, x_2, ..., x_n) \mapsto \sum_{i=1}^n \lambda_i x_i$. The convex envelope C of $\bigcup_{i=1}^n A_i$ is the image of $B \times \prod_{i=1}^n A_i$ under this mapping; as $B \times \prod_{i=1}^n A_i$ is compact and E is Hausdorff, it follows that C is compact.