For some $\delta>0$, let $X$ and $X_\delta$ be two bounded convex polytopes in $\mathbb{R}^n$, defined as $X = \{x \in \mathbb{R}^n : Ax \leq b \}$ and $X_\delta = \{x \in \mathbb{R}^n : Ax \leq b + \delta u\}$ respectively, where $A \in \mathbb{R}^{m\times n}$ (assume $m\geq n$ and $\mathrm{Rank} A = n$), $b \in \mathbb{R}^m$ and $u = (1,\ldots,1)^\mathrm{T} \in \mathbb{R}^m$.
I would like to have an estimate on their Hausdorff distance $d_H(X,X_\delta)$, that is, upper/lower bounds that are fairly easy to compute.
What do you suggest?
Here is an (incomplete) idea: Since $X\subseteq X_\delta$, $d_H(X,X_\delta) = \inf \{ \varepsilon \geq 0: X_\delta \subseteq X \oplus \varepsilon \mathcal{B}\}$, where $\mathcal{B}$ is the closed unit ball (we work with the Euclidean norm, $||\cdot||_2$). Now (*) take $x' \in \partial X_\delta := \{x \in \mathbb{R}^n : Ax' = b + \delta u \}$ and $x \in \partial X := \{x \in \mathbb{R}^n : Ax = b \}$. It follows that $A(x'-x) = \varepsilon u$; denoting $x'-x = \varepsilon d$ for some $d \in \mathcal{B}$, we get $Ad = u\dfrac{\delta}{\varepsilon}$.
From the SVD of $A$ we get that $\sigma_{min}(A) \leq ||u||_2 \dfrac{\delta}{\varepsilon} \leq \sigma_{max}(A)$, and noting that $||u||_2 = \sqrt{m}$, the upper bound $\varepsilon \leq \dfrac{\delta \sqrt{m}}{\sigma_{min}(A)}$ and the lower bound $\varepsilon \geq \dfrac{\delta \sqrt{m}}{\sigma_{max}(A)}$ follow.
(*) Drawback: is it justified to consider only the points in the faces? (for me it is intuitive if I draw it, but how do you go on to formalizing this?)