Proposition 12

— If A is a convex set in E, then the convex cone generated by A is identical with $C = \bigcup_{\lambda > 0} \lambda A$.

The set C is clearly a cone; it is sufficient to show that C is convex. Let $\lambda x, \mu y$ be two points of C ($\lambda > 0, \mu > 0, x \in A, y \in A$). Let $\alpha, \beta$ be two numbers $> 0$ such that $\alpha + \beta = 1$. Then $\alpha \lambda x + \beta \mu y = (\alpha \lambda + \beta \mu) z$, with $z \in A$, and $\alpha \lambda + \beta \mu > 0$; hence $\alpha \lambda x + \beta \mu y \in C$.