Exercise 18, § 2

Let C be a proper closed convex cone of vertex 0 in $\mathbf{R}^n$. Show that the complement of the set $C \cap S_{n-1}$ on the sphere $S_{n-1}$ is homeomorphic to $\mathbf{R}^{n-1}$ (make a stereographic projection from a point of $C \cap S_{n-1}$, and use exerc. 12 of GT, VI, § 2). If C contains an interior point, show that $C \cap S_{n-1}$, is homeomorphic to the closed ball $B_{n-1}$ (same method).