Corollary 1

If C is a non-empty convex cone, the vector space generated by C is the set $C - C$ (the set of points $x - y$ where x, y vary in C).

For, if $V = C - C$, then V is non empty, we have $\lambda V = V$ for all $\lambda \neq 0$, and $V + V = C + C - (C + C) \subset C - C = V$, which shows that V is a vector subspace. Finally every vector subspace that contains C also contains V.