Proof of the Honeycomb Asymptotics for Optimal Cheeger Clusters

We prove that, in the limit as $k\to +\mathrm{\infty }$, the hexagonal honeycomb solves the optimal partition problem in which the criterion is minimizing the largest among the Cheeger constants of $k$ mutually disjoint cells in a planar domain. As a by-product, the same result holds true when the Cheeger constant is replaced by the first Robin eigenvalue of the Laplacian.