Shape optimization with inradius constraint and emergence of honeycomb structures

We maximize the average torsional rigidity among open subsets of ${\mathbb{R}}^2$ with fixed inradius whose complements consist of pairwise disjoint balls of fixed radius $\epsilon$ with mutual distances bounded below. We prove that, provided $\epsilon$ is sufficiently small, the optimal value is asymptotically attained by sequences of domains whose complements exhibit a regular honeycomb structure. The proof is reduced, via Delaunay triangulations, to the analysis of a mixed Dirichlet–Neumann problem on triangular domains with circular cutouts at their vertices. Our approach is quite general and extends to other variational energies, such as the Cheeger constant, for which we also derive an explicit bound of the ratio $\epsilon \mathrm{/}$ inradius.