Given an open, bounded set
$\Omega $
in
$\mathbb {R}^N$
, we consider the minimization of the anisotropic Cheeger constant
$h_K(\Omega )$
with respect to the anisotropy K, under a volume constraint on the associated unit ball. In the planar case, under the assumption that K is a convex, centrally symmetric body, we prove the existence of a minimizer. Moreover, if
$\Omega $
is a ball, we show that the optimal anisotropy K is not a ball and that, among all regular polygons, the square provides the minimal value.