Corollary 2

— The interior $\dot{C}$ of a convex cone C, is itself a convex cone; if $\dot{C}$ is not empty then it coincides with the interior of $\overline{C}$, and $\overline{C}$ is a pointed convex cone that coincides with the closure of $\dot{C}$.

Since homotheties of ratio $> 0$ and centre 0 transform C into itself, they do the same for $\dot{C}$, thus $\dot{C}$ is a cone; the remainder of the corollary follows from cor. 1 and the obvious remark that if C is not empty then $\overline{C}$ contains the vertex of C.

Let H be a closed hyperplane in the topological vector space E over $\mathbf{R}$; it has an equation of the form $f(x) = \alpha$, where $f$ is a continuous linear form that is not identically zero in E (I, p. 13, th. 1). The closed half spaces defined respectively by $f(x) \leqslant \alpha$ and $f(x) \geqslant \alpha$ are therefore closed convex sets; their complements defined respectively by $f(x) > \alpha$ and $f(x) < \alpha$, are open convex sets. We say that these half-spaces are the closed (resp. open) half spaces determined by H.