Optimization of the anisotropic Cheeger constant with respect to the anisotropy

Given an open, bounded set $\mathrm{\Omega }$ in ${\mathbb{R}}^N$, we consider the minimization of the anisotropic Cheeger constant $h_K(\mathrm{\Omega })$ 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 $\mathrm{\Omega }$ 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.