Proposition 9

A pointed convex cone C is proper if and only if the non-pointed cone $C'$, which is the complement of 0 in C, is convex.

If C contains a line through 0 then clearly $C'$ is not convex. Suppose now that C is proper and let x, y be two points of $C'$. The closed segment with end points x, y is contained in C; if it contains 0 then $\lambda x + (1 - \lambda) y = 0$ for some $\lambda$ with $0 < \lambda < 1$, therefore $x = \mu y$ with $\mu < 0$. Thus C contains the line through 0 and x, contrary to hypothesis.