For a finite-dimensional normed space V and a subset X with finite Hausdorff distance from V, we prove that the Gromov--Hausdorff distance between X and V is at least the Hausdorff distance between X
Nearby in the stack
and
V
, divided by twice the relative Jung constant of
V
. If
V
furthermore satisfies a certain intersection property, we show a stronger result where the relative Jung constant can be replaced with its absolute version. Key words: Normed spaces, Jung constant, Hausdorff distance, Gromov--Hausdorff distance.