Convex Cone - Half-spaces

Half-spaces

A (linear) hyperplane of V is a maximal proper linear subspace of V. An open (resp. closed) half-space of V is any subset H of V defined by the condition L(x) > 0 (resp. L(x)0), where L is any linear function from V to its scalar field. The hyperplane defined by L(v) = 0 is the bounding hyperplane of H.

Half-spaces (open or closed) are convex cones. Moreover, any convex cone C that is not the whole space V must be contained in some closed half-space H of V. In fact, a topologically closed convex cone is the intersection of all closed half-spaces that contain it. The analogous result holds for any topologically open convex cone.