Definition 3

A subset C of an affine space E is a cone with vertex x0 if C is invariant for all homotheties of centre x_0 and ratio > 0.*

We shall suppose in this No. and in the one following, that we have chosen the vertex of the cone being considered, as origin in E; i.e. we suppose that E is a vector space, and when we speak of a cone, it is to be understood that this cone has vertex 0. The set of points of the form $\lambda a$ for $\lambda > 0$ (resp. $\lambda \geq 0$), where $a$ is a non-null vector, is called an open half line (resp. closed half-line) originating at 0.

A cone C of vertex 0 is said to be pointed if $0 \in C$, and non-pointed otherwise. A pointed cone is either the single point $\{0\}$ or is the union of a set of closed half-lines originating at 0. A non-pointed cone is the union (possibly empty) of open half lines originating at 0. If C is a non-pointed cone, then $C \cup \{0\}$ is a pointed cone. If C is a pointed cone, then $C - \{0\}$ is a non-pointed cone.

If C is a non-pointed convex cone, then $C \cup \{0\}$ is a pointed convex cone. However, if C is a pointed convex cone, $C - \{0\}$ is not necessarily convex. We say that a pointed convex cone is proper if it does not contain any line passing through 0. Then