Let $M$ be a metric space. Then the *doubling dimension* of $M$, denoted $\dim M$, is defined to be the minimum value $k$ such that every ball in $M$ of radius $r$ can be covered by at most $2^k$ balls of radius $r/2$.

In the paper "Small Hop-diameter Sparse Spanners for Doubling Metrics" by Chan/Gupta, the authors write:

Apart from being a generalization of the $\ell_p$ notion of dimension, designing algorithms that only use the doubling properties (instead of the geometry of $\mathbb R^k$) has other advantages: the notion of doubling dimension is fairly resistant to small perturbations in the distances:

for instance, if one takes a distance matrix of a set of points in $\ell_p^k$ and slightly changes some of the entries, then the doubling dimension does not change by much,but the metric may not remain isometrically embeddable in $\ell_p$ (into any number of dimensions).

I have two related questions about this statement.

Is there a known formalization of this stability, perhaps with respect to the Hausdorff distance on (finite) subsets of $\ell_p^k$? That is, do we have something like the following: Let $A$ and $B$ be finite subsets of $\ell_p^k$. Then $\lvert\dim A - \dim B\rvert \leq d_\text H(A,B)$, where $d_\text H$ is the Hausdorff distance?

Does this stability generalize to arbitrary finite metric spaces, with respect to, e.g., the Gromov–Hausdorff distance? That is, if $A$ and $B$ are finite metric spaces, do we have something like $\lvert\dim A - \dim B\rvert \leq d_\text{GH}(A,B)$, where $d_\text{GH}$ is the Gromov–Hausdorff distance?

If 'No' to the above, are there any sensible formalizations of Chan & Gupta's statement that make their claim of stability precise?