Optimization of the anisotropic Cheeger constant with respect to the anisotropy

Abstract

Abstract 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.