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**Inserted:** 1 jul 2022

**Last Updated:** 18 feb 2023

**Journal:** Canad. Math. Bull.

**Year:** 2023

**Doi:** 10.4153/S0008439523000152

**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.

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