This is an extract from a linear optimization book:
However, I do not understand why both of these conditions describe the same polyhedra? The linear equation results in a set $$\left\{\begin{bmatrix}x\\y\end{bmatrix} \in \mathbb{R}^2 : -2x - y \leq 0 \text{ and } 2x - y \leq 0 \right\}$$ which is a line of slope 1/2:
The cone results in a set which covers all of the points between the hands of the function $y(x) = |2x|$:
Can it be that this is a typo?
$$\left\{\begin{bmatrix}x\\y\end{bmatrix} \in \mathbb{R}^2 : -2x - y \leq 0 \text{ and } 2x - y \leq 0 \right\}$$ is not a line.
We have $-2x \le y$ and $ 2x\le y$. It is the intersection of the two halfspaces.
It can be written as $y \ge \max\{-2x, 2x\}=2|x|$.