Exercise 25, § 7

*a) Let $C \subset \mathbf{R}^n$ be a closed convex cone with vertex 0. Show that the projections of $C$ on every 2-dimensional subspace of $\mathbf{R}^n$ are closed if and only if $C$ is a polyhedral cone (exerc. 24). (Reduce to the case when $C$ contains no line : argue by induction on $n$ using the existence of a compact sole $S$ of $C$ (II, p. 90, exerc. 21, a)), and project onto a hyperplane parallel to a line joining 0 to an extremal point of $S$, and deduce that $S$ is locally polyhedral (exerc. 24). *b) Deduce from a) that, if we give $\mathbf{R}^n$ the order for which $C$ is the set of elements $\geqslant 0$, then, every positive linear form on any vector subspace $F$ of $\mathbf{R}^n$ can be extended to a positive linear form on $\mathbf{R}^n$ if, and only if, $C$ is a polyhedral cone (apply a) to the polar cone $C^\circ$. If $C$ is the cone in $\mathbf{R}^3$ generated by the $(\xi_1, \xi_2, \xi_3)$ such that $\xi_1 = 1, \xi_3 \geqslant (\xi_2^3)^-$, and $F$ is the subspace $\xi_3 = 0$ give an example of a positive linear form on $F$ that cannot be extended to a positive linear form on $\mathbf{R}^3$. *c) Let $A$ be a polyhedron in $\mathbf{R}^n$. Then, the convex envelope of $A \cup B$ is closed for every polyhedron $B$, if and only if, $A$ is compact (use exerc. 24, d)).