Suppose we have a nonempty polytope given by $P=\{x \in \mathbb{R}^n_{\geq 0}:Ax \leq b\}$. Does there exist an $\epsilon>0$ such that $P'=\{x \in \mathbb{R}^n_{\geq 0}:Ax \leq b - \overline{\epsilon}\}$ is nonempty, where $\overline{\epsilon}$ is the vector whose components are all $\epsilon$?
This seems reasonable geometrically, but I'm not sure how to prove it analytically. I can also imagine that there might be some edge cases where this isn't possible.
For a simple counterexample, consider $n=m=1$, $A=(1)$, and $b=0$. Then $P=\{0\}$ is nonempty, but $P'$ is empty for all $\epsilon>0$.