Proposition 7

— Let A and B be two convex subsets of the vector space E. For any real numbers $\alpha, \beta$ the set $\alpha A + \beta B$ (set of points of the form $\alpha x + \beta y$, where $x$ varies in A, and $y$ in B*) is convex*.

For $\alpha A + \beta B$ is the image of the convex subset $A \times B$ of $E \times E$ under the linear mapping $(x, y) \mapsto \alpha x + \beta y$ of $E \times E$ in $E$.