Pointedness, Connectedness, and Convergence Results in the Space of Closed Convex Cones

Let $\mathcal{K}(H)$ denote the collection of all closed convex cones in a finite dimensional real Hilbert space H. Painleve-Kuratowski convergence in this space has been the subject of numerous publications. However, the link between convergence and pointedness has not been explored in its full extent. This note contributes to close this gap. Some results concerning the connected components of $\mathcal{K}(H)$ are also considered.