SCHWEITZER 1981 Problem 5

Let $K$ be a convex cone in the $n$ -dimensional real vector space $\mathbb{R}^n$ , and consider the sets $A=K \cup (-K)$ and $B=(\mathbb{R}^n \setminus A) \cup \{ 0 \}$ ( $0$ is the origin). Show that one can find two subspaces in $\mathbb{R}^n$ such that together they span $\mathbb{R}^n$ , and one of them lies in $A$ and the other lies in $B$ .

J. Szucs