PreprintarXiv licencearXiv
Shape optimization with inradius constraint and emergence of honeycomb structures
Abstract
We maximize the average torsional rigidity among open subsets of ℝ2 with fixed inradius whose complements consist of pairwise disjoint balls of fixed radius ε with mutual distances bounded below. We prove that, provided ε 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 ε/ inradius.
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BibTeX
@misc{bucur2026shape,
title = {{Shape optimization with inradius constraint and emergence of honeycomb structures}},
author = {Dorin Bucur and Giuseppe Buttazzo and Alexis de Villeroché},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08072},
eprint = {2610.08072},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08072},
}
RIS
TY - UNPB
TI - Shape optimization with inradius constraint and emergence of honeycomb structures
AU - Dorin Bucur
AU - Giuseppe Buttazzo
AU - Alexis de Villeroché
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08072
UR - https://arxiv.org/abs/2610.08072
AB - We maximize the average torsional rigidity among open subsets of ℝ^2 with fixed inradius whose complements consist of pairwise disjoint balls of fixed radius ε with mutual distances bounded below. We prove that, provided ε 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 ε/ inradius.
ER -