Proposition 10

A subset C of E is a convex cone if and only if $C + C \subset C$ and $\lambda C \subset C$ for all $\lambda > 0$.

For the condition $\lambda C \subset C$ for all $\lambda > 0$ characterises the cones. If C is convex we have $C + C = \frac{1}{2}C + \frac{1}{2}C = C$ (II, p. 8, Remark). Conversely, if the cone C is such that $C + C \subset C$, then for $0 < \lambda < 1$, we have $\lambda C + (1 - \lambda) C = C + C \subset C$, which shows that C is convex.