Exercise 22, § 5

In $\mathbf{R}^n$ let $(C_i)_{1 \leq i \leq m}$ be a finite family of closed convex cones with vertex 0, such that the sum of any $n$ of them is distinct from $\mathbf{R}^n$. Show that there exists a hyperplane H, passing through 0, such that, for any index i no pair of points of $C_i$ are strictly separated by H. (Distinguish two cases :

α) Either there exists a number $r < n$ and $r$ indices, say, 1, 2, ..., $r$ such that $C_i$ for $i \leq r$ generates a cone which contains a vector subspace $V$ of dimension $\geq r$. Argue by induction on $n$, projecting on the orthogonal to $V$.

β) Or, for all $r < n$, any $r$ of the $C_i$ generate a cone $C$ such that the maximal vector subspace $C \cap (-C)$ contained in $C$ is of dimension $< r$. Consider then a set of the cones $C_i$, maximal with respect to being contained in a half-space; there are at least $n$ cones in a maximal set. Let $\Gamma$ be the cone generated by the union of the cones belonging to this maximal set. If $C_j$ is a cone which does not belong to the maximal set considered, show that $C_i \subset -\Gamma$. For this, argue by reductio ad absurdum, showing that in the contrary case there exists a frontier point of $-\Gamma$ (relative to the vector subspace generated by $\Gamma$) that is interior to $C_j$ (relative to the vector subspace generated by $C_j$). Write such a point as the sum of the least number $s$ of vectors, of which each belongs to a cone $-C_i$, among those $C_i$ used in defining $\Gamma$; then $s \leq n-1$. Prove finally that these cones and $C_j$ generate a convex cone containing an $s+1$ dimensional vector subspace, contradicting the hypothesis; for this use exerc. 4 of II, p. 65.)