I am reading a proof which states without proof that:
If a polyhedron $P$ in $\mathbb{R}^n$ has dimension dim($P$) $\geq 1$ then there exists $c\in\mathbb{R}^n$ such that max$\{c^Tx\text{ }|\text{ }x\in P\} - \text{min}\{c^Tx\text{ }|\text{ }x\in P\} = 1$.
Apparently this is supposed to be straightforward, since no proof is given, but I do not know why this is true. Would someone be able to help clear this up for me?
If you can get $\max(c^Tx) - \min(c^Tx) > 0$ for some $c$, then scaling $c$ suitably gives what you want.