Show that there are $A,B$ with $\mathcal{P}:=\{x\in\mathbb{R}^n:Ax≤\mathbf{1},\;Bx≤\mathbf{0} \}$

Question

$\mathcal{P}$ is a polyhedron, such that $\mathbf{0}\in\mathcal{P}$. Show that there are Matrices $A,B$ with:
$$\mathcal{P}=\{x\in\mathbb{R}^n:Ax≤\mathbf{1},\;Bx≤\mathbf{0} \}$$There seems to be so little to go with, that I don't really know how to go about answering this question. I would appreciate someone helping me with this, either through hints or a full solution.

Answer 1

I suppose that a polyhedron is the set of points satisfying an inequality of the type $$ Cx \le b $$ where $C$ is a matrix, $b$ a vector. Let $c_i$ be the $i$-th row of $C$.

Since $0\in P$, it follows $b_i \ge0$. Now if $b_i=0$, then put the inequality $c_i^Tx \le0$ into matrix $B$. If $b_i>0$, then put the inequality $c_i^Tx\le b_i$ in the form $(b_i^{-1}c_i^T)x \le 1$ into matrix $A$.