# Lipschitz-continuity of convex polytopes under the Hausdorff metric

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!

• It might be helpful if you give a more intuitive, in words, summary of what the continuity result means. Commented Dec 2, 2021 at 13:41
• @SamHopkins Thank you for your comment, I think that's a great idea! I just edited my question accordingly. Commented Dec 2, 2021 at 17:23