Given the polyhedron $$P = \{v \in \mathbb R^2 \mid Av \le b\}$$ with $$A = \begin{bmatrix} -1 & -1 \ 2 & -1 \ -1 & 2 \ 1 & 2 \end{bmatrix}$$ and $$b = \begin{bmatrix} 0 \ 1 \ 1 \ 2 \end{bmatrix}$$, I want to find the edges of P.

I know that if F is a non-empty face then $$F=P_I$$ for some $$I \subset \{1,2,3,4\}$$, where $$P_I = \{x \in P \mid A_I x = b_I\}$$ Also, if there exist $$z \in P_I$$ such that $$I(z) =\{i \in I \mid a_i z = b_i \} = I$$ then $$\text {dim} P_I = d - \text {rk} A_I$$
How can I find the edges of P using this knowledge?

Question

Given the polyhedron $$P = \{v \in \mathbb R^2 \mid Av \le b\}$$ with $$A = \begin{bmatrix} -1 & -1 \ 2 & -1 \ -1 & 2 \ 1 & 2 \end{bmatrix}$$ and $$b = \begin{bmatrix} 0  \ 1 \ 1  \ 2 \end{bmatrix}$$, I want to find the edges of P.

I know that if F is a non-empty face then $$F=P_I$$ for some $$I \subset \{1,2,3,4\}$$, where $$P_I = \{x \in P \mid A_I x = b_I\}$$ Also, if there exist $$z \in P_I$$ such that $$I(z) =\{i \in I \mid a_i z = b_i \} = I$$ then $$\text {dim} P_I = d - \text {rk} A_I$$
How can I find the edges of P using this knowledge?

anonymous · Answer

Note. I am specifically asked to use the the proposition stated above, therefore it is not enough to simply draw the Polyhedron from the lines and then find the edges.

(This is undergraduate convexity, dont know if this is something you've learned about @ganeshie8 , but you've helped me with some similar stuff before)

thomas5267 · Answer

Leaving my name here for receiving updates.  I have no idea how to do this.

ganeshie8 · Answer

@Kainui