Exercise 7, § 6

a) Let F, G be two vector spaces in separating duality and C a weakly closed convex cone in F. Let M be a finite dimensional vector subspace of G. Show that, either there exists $y_0 \in C$ such that $y_0 \in M^\circ$ and $y_0 \neq 0$, or there exists $z_0 \in M$ such that $z_0 \in C^\circ$ and $z_0 \neq 0$ (argue by induction on the dimension of M). If C does not contain any line and if the two preceding properties are simultaneously satisfied, show that $z_0$ cannot be an internal point of $C^\circ$. b) Let the two matrices $(a_{ij}), (b_{ij})$ with real entries in $n$ rows and $m$ columns, be such that $a_{ij} > 0$ for every pair $(i, j)$. Show that there is a unique value of $\lambda \in \mathbf{R}$ such that there are two vectors $x = (x_j) \in \mathbf{R}^m, y = (y_i) \in \mathbf{R}^n$ satisfying the relations $x \neq 0,\ y \neq 0,\ x_j \geq 0,\ y_i \geq 0$ for all $i, j$ and finally such that

(Putting $c_{ij} = \lambda a_{ij} - b_{ij}$ for $1 \leq i \leq n,\ 1 \leq j \leq m$, and $c_{n+i,j} = \delta_{ij}$ (Kronecker’s index) for $1 \leq i \leq m$ show that the problem reduces to finding a vector $x \in \mathbf{R}^m$ and a vector $z = (z_i) \in \mathbf{R}^{n+m}$ which are non null and satisfy the relations

(3)

(4) and $z_i \geq 0$ for $1 \leq i \leq n + m$. Remark that, if (3) has a solution for one value $\lambda_0$ of $\lambda$, then it also has a solution for $\lambda \geq \lambda_0$, and that if (4) has a solution for $\lambda_0$, then it also has a solution for $\lambda \leq \lambda_0$. Finally use a).