  The regular $N$-gon provides the minimal Cheeger constant in the class of all
$N$-gons with fixed volume. This result is due to a work of Bucur and Fragal\`a
in 2014. In this note, we address the stability of their result in terms of the
$L^1$ distance between sets. Furthermore, we provide a stability inequality in
terms of the Hausdorff distance between the boundaries of sets in the class of
polygons having uniformly bounded diameter. Finally, we show that our results
are sharp, both in the exponent of decay and in the notion of distance between
sets.
