Exercise 2, § 6

In $\mathbf{R}^n$, let $(C_i)_{1 \leq i \leq m}$ be a family of $m \geq n + 1$ convex cones with vertex 0; show that if, for any $n + 1$ of these cones there exists a hyperplane H through 0 and such that the cones lie on the same side of H, then there exists a hyperplane $H_0$ such that all the cones $C_i (1 \leq i \leq m)$ lie on the same side of $H_0$ (cf. II, p. 68, exerc. 21, a)).