Recently, I proved the following Lipschitz-continuity like result for convex polytopes:

Let $A\in\mathbb R^{m\times n}$ and $b,b'\in\mathbb R^m$ be given such that $\{x\,:\,Ax\leq 0\}=\{0\}$ (which is equivalent to boundedness of all induced polytopes) and that $\{x\in\mathbb R^n\,:\,Ax\leq b\},\{x\in\mathbb R^n\,:\,Ax\leq b'\}$ are non-empty. Then $$ \delta\big(\{x\in\mathbb R^n\,:\,Ax\leq b\},\{x\in\mathbb R^n\,:\,Ax\leq b'\}\big)\leq \Big( \max_{A_0\in\operatorname{GL}(n,\mathbb R),A_0\subset A}\|A_0^{-1}\| \Big)\|b-b'\|_1 $$ where the operator norm $\|\cdot\|$ as well as the Hausdorff metric $\delta$ are taken with respect to $(\mathbb R^n,\|\cdot\|_1)$. Also $A_0\subset A$ is short for "every row of $A_0$ is also a row of $A$" so the above maximum is taken over all invertible submatrices of $A$.

What this intuitively means is that if two polytopes have parallel faces (i.e. they are both described by the same $A$ matrix), but the *location* of these faces differs ($b\neq b'$), then the distance between the polytopes is upper bounded by the distance between the vectors $b,b'$ times a "geometrical" constant coming from $A$.

Hence the function $b\mapsto\{x\in\mathbb R^n:Ax\leq b\}$ (with suitable domain such that the co-domain equals all non-empty subsets of $\mathbb R^n$) is Lipschitz continuous with a constant determined by $A$.

This came up as a lemma to something only vaguely related which is why I don't have a problem with posting it publicly. Actually if this was a known result then my manuscript could be shortened by 3 pages. Thus my quesiton is:

Is the above result known and, if so, where can in be found in the (convex polytope-)literature?

I would be surprised if nobody has thought about this until now. While I haven't seen this result in the books of Grünbaum and Schrijver or the few papers on convex polytopes I am aware of, this is not the field I usually work in; hence this might very well be known but beyond my mathematical horizon. Thanks in advance for any answer or comment!