Exercise 1, § 5

a) Let E be a vector space. We say that a pointed convex cone C (of vertex 0) in E is maximal if C is a maximal element of the set of convex pointed cones of vertex 0 and $\neq E$, ordered by inclusion. Show that a pointed convex cone C is maximal if, and only if, it is a closed half-space defined by a hyperplane which passes through 0. To establish this result, prove successively the following properties of a maximal pointed convex cone C ;

α) We have $C \cup (-C) = E$ (argue by obtaining a contradiction).

β) If $z$ is a non-internal (II, p. 26) point of $C$ then $-z \in C$ (same method). Deduce that $C$ contains internal points.

γ) The largest vector subspace $H = C \cap (-C)$ contained in $C$ is a hyperplane. (Passing to the quotient space $F = E/H$, this reduces to demonstrating, using β) that if all the points of $C$ other than the vertex are internal, then $E$ is necessarily of dimension 1.)

b) Give an example of a maximal non-pointed convex cone (in the set of non-pointed convex cones of vertex 0) which has no internal point (cf. II, p. 65, exerc. 5).