Definition 3

Let C be a convex set in a Hausdorff topological vector space E. A compact convex non-empty set A of C is called a cap of C if the complement C − A of A in C is convex.

Let C be a pointed convex cone with vertex 0 in E and let A be a cap of C. Write B = C − A. For every closed half-line L ⊂ C originating at 0, the sets L ∩ A and L ∩ B are convex sets that are complements in L, whose union is L, and such that L ∩ A is compact. As L ∩ A is non-empty for at least one half-line L, we see that 0 ∈ A, thus L ∩ A is a closed segment with an end point at 0. If A exists then C is proper.