  In this paper we introduce a pruning of the medial axis called the
$(\lambda,\alpha)$-medial axis ($\textrm{ax}_\lambda^\alpha $). We prove that
the $(\lambda,\alpha)$-medial axis of a set $K$ is stable in a Gromov-Hausdorff
sense under weak assumptions. More formally we prove that if $K$ and $K'$ are
close in the Hausdorff ($d_H$) sense then the $(\lambda,\alpha)$-medial axes of
$K$ and $K'$ are close as metric spaces, that is the Gromov-Hausdorff distance
($d_{GH}$) between the two is $\frac{1}{4}$-H{\"o}lder in the sense that
$d_{GH} (\textrm{ax}_\lambda^\alpha (K),\textrm{ax}_\lambda^\alpha (K'))
\lesssim d_H(K,K')^{1/4}$. The Hausdorff distance between the two medial axes
is also bounded, by $d_{H} (\textrm{ax}_\lambda^\alpha
(K),\textrm{ax}_\lambda^\alpha (K')) \lesssim d_H(K,K')^{1/2}$. These
quantified stability results provide guarantees for practical computations of
medial axes from approximations. Moreover, they provide key ingredients for
studying the computability of the medial axis in the context of computable
analysis.
