Generating a point in a rational polytope $P \subseteq R^k$ given a point in $P^\epsilon$
From any choice of a polytope $P$ in ${\mathbb R}^k$, $\epsilon$, and a point $q$ in ${\mathbb R}^k$ it is possible to find a polytope $\hat P$ in ${\mathbb R}^{k+1}$, together with an embedding of ${\mathbb R}^k$ into ${\mathbb R}^{k+1}$, such that $\hat P$ is within $\epsilon$ Hausdorff distance of (the embedded image of) $P$ and such that (the embedded image of) $q$ belongs to $\hat P^\epsilon$. To do this, simply make the facets of $\hat P$ be nearly parallel to the embedded image of ${\mathbb R}^k$, so that translating them by $\epsilon$ in ${\mathbb R}^{k+1}$ causes their intersection with ${\mathbb R}^k$ to move away from $\hat P$ by a much greater distance.

Because $q$ was arbitrary, the knowledge of $q$ is of no use in finding a point in or near $P$; everything you could do with it you could do without it. But, because $\hat P$ and $P$ are so close, finding a point near $P$ is equivalent to finding a point near $\hat P$. Therefore, the knowledge of $q$ (a point in $\hat P^\epsilon$) is of no use in finding a point near $\hat P$.