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



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> 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 $A$.
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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:



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



Lipschitz-continuity of convex polytopes under the Hausdorff metric
This is a classic question in the literature on linear programming, since it is related to the stability of the feasible set (and hence the solutions) under perturbation of the parameters.


The classic work in this field is:


* A. J. Hoffman, Approximate solutions of systems of linear inequalities, *J. Res. Nat. Bur. Standards*, 1952.


A more recent work, which essentially states your result as a special case (see Theorem 2.4 and the Lipschitz constant on p.19), is:


* W. Li, The sharp Lifshitz constants for feasible and optimal solutions of a perturbed linear program, *Linear Algebra and Its Applications*, 1993.


