Remark

— 1) With the hypotheses of prop. 12, if $0 \notin A$, then the cone C is non-pointed, thus $C \cup \{0\}$ is proper.

2) Let A be any convex set in E; consider the convex set $A_1 = A \times \{1\}$ in the space $F = E \times \mathbf{R}$ and the convex cone C with vertex 0 that is generated by $A_1$. Prop. 12 shows that $A_1$ is the intersection of C and of the hyperplane $E \times \{1\}$ in F. Every convex set in E can, therefore, be considered as the projection on E of the intersection of a convex cone with vertex 0 in F and the hyperplane $E \times \{1\}$.