On the honeycomb conjecture for a class of minimal convex partitions
Authors:
Dorin Bucur, Ilaria Fragalà, Bozhidar Velichkov and Gianmaria Verzini
Journal:
Trans. Amer. Math. Soc. 370 (2018), 7149-7179
MSC (2010):
Primary 52C20, 51M16, 65N25, 49Q10
DOI:
https://doi.org/10.1090/tran/7326
Published electronically:
May 17, 2018
MathSciNet review:
3841845
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We prove that the planar hexagonal honeycomb is asymptotically optimal for a large class of optimal partition problems, in which the cells are assumed to be convex, and the criterion is to minimize either the sum or the maximum among the energies of the cells, the cost being a shape functional which satisfies a few assumptions. They are: monotonicity under inclusions; homogeneity under dilations; a Faber-Krahn inequality for convex hexagons; a convexity-type inequality for the map which associates with every $n \in \mathbb {N}$ the minimizers of $F$ among convex $n$-gons with given area. In particular, our result allows us to obtain the honeycomb conjecture for the Cheeger constant and for the logarithmic capacity (still assuming the cells to be convex). Moreover, we show that, in order to get the conjecture also for the first Dirichlet eigenvalue of the Laplacian, it is sufficient to establish some facts about the behaviour of $\lambda _1$ among convex pentagons, hexagons, and heptagons with prescribed area.
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Additional Information
Dorin Bucur
Affiliation:
Institut Universitaire de France , Laboratoire de Mathématiques UMR 5127 , Université de Savoie, Campus Scientifique , 73376 Le-Bourget-Du-Lac, France
MR Author ID:
349634
Email:
dorin.bucur@univ-savoie.fr
Ilaria Fragalà
Affiliation:
Dipartimento di Matematica , Politecnico di Milano , Piazza Leonardo da Vinci, 32 , 20133 Milano, Italy
MR Author ID:
629098
Email:
ilaria.fragala@polimi.it
Bozhidar Velichkov
Affiliation:
Laboratoire Jean Kuntzmann (LJK), Université Grenoble Alpes, Bâtiment IMAG, 700 Avenue Centrale, 38401 Saint-Martin-d’Hères, France
MR Author ID:
1000813
Email:
bozhidar.velichkov@univ-grenoble-alpes.fr
Gianmaria Verzini
Affiliation:
Dipartimento di Matematica , Politecnico di Milano , Piazza Leonardo da Vinci, 32 , 20133 Milano, Italy
MR Author ID:
667514
Email:
gianmaria.verzini@polimi.it
Keywords:
Optimal partitions,
honeycomb conjecture,
Cheeger constant,
logarithmic capacity,
discrete Faber-Krahn inequality.
Received by editor(s):
March 10, 2017
Published electronically:
May 17, 2018
Additional Notes:
This work was partially supported by the PRIN-2015KB9WPT Grant: “Variational methods, with applications to problems in mathematical physics and geometry”, by the ERC Advanced Grant 2013 n. 339958: “Complex Patterns for Strongly Interacting Dynamical Systems - COMPAT”, and by the INDAM-GNAMPA group.
Article copyright:
© Copyright 2018
American Mathematical Society