I was trying to build a polytope using implicit inequalities, one of the examples I found was the following
$$P=\{(x, y, z): x+y+z \leq 4, x+y \geq 3, z \leq 1, 1 \leq y \leq 2 \}$$
From this we have that $x+y+z \leq 4$, $x+y \geq 3$ and $z \leq 1$ are the implicit equalities in the system of inequalities that defining P.
My question is, graphically is it possible to obtain this polytope? I have tried Wolfram|Alpha but it has not been possible, any suggestions?
Remember that...
We say that $a^{i}x \leq b_{i}$ is an implicit equality of $Ax \leq b$ if an H-polytope $P$ is contained in the hyperplane $\{x\in \mathbb{R}^{d}: a^{i}x=b_{i} \}$.
- We denote by $A^{=}x \leq b^{=}$ the system containing all implicit equalities of $Ax \leq b$.
- We denote by $A^{\le }x \leq b^{ \le}$ the system containing all remaining inequalities of $Ax \leq b$.
Additionally, do you know of any example of a polytope that is described by its implicit equalities?