Optimization of the anisotropic Cheeger constant with respect to the anisotropy

Given an open, bounded set Ω in R^n, we consider the minimization of the anisotropic Cheeger constant h_K (Ω) 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 Ω 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.