Proposition 3

In a vector space E, let C be a convex cone with vertex 0, and let x₀ ≠ 0 be a point of C, and D a half-line that is contained in C, originating from 0 and containing x₀. Let H be a hyperplane containing x₀ and not passing through 0. Then D is an extremal generator of C if and only if x₀ is an extremal point of H ∩ C.

The condition is clearly necessary. Conversely, suppose that it is satisfied; suppose that there is a line D' not containing D, passing through x₀ and such that D' ∩ C contains an open segment to which x₀ belongs. Let y ≠ 0 be a direction vector of D'; the hypotheses imply that the point (1 + λ)x₀ + μy belongs to C for |λ| and |μ| sufficiently small. But then, in the plane P determined by D and D' and carrying the canonical topology, x₀ is an interior point of P ∩ C, and it follows that the line P ∩ H contains an open segment contained in H ∩ C and to which x₀ belongs. This contradicts the hypothesis.