On the honeycomb conjecture for a class of minimal convex partitions

Abstract

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 ∈ N the minimizers of F among convex n-gons with given area. In particular, our result allows 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 λ1 among convex pentagons, hexagons, and heptagons with prescribed area.

Proof of the honeycomb asymptotics for optimal Cheeger clusters

“…Finally, relying on Theorem 2 and using a blow-up argument, we obtain that the honeycomb conjecture for the Cheeger constant holds true for every Lipschitz domain Ω in the following asymptotic form (which is exactly the same as in [3], without the convexity assumption on the cells): , β). Precisely, given β > 0 (fixed), λ 1 (Ω j , β) is the lowest positive number for which the equation…”

“…More recently, problems of the kind (1) and (3) have been studied in [3], where it is shown that, under the a priori requirement that all the cells of the partitions are convex, the honeycomb conjecture holds true under the form (4) lim k→+∞ |Ω| 1/2 k 1/2 M k (Ω) = h(H) , lim k→+∞ |Ω| 1/2 k 3/2 m k (Ω) = h(H) , where H denotes the unit area regular hexagon. Clearly, the convexity assumption made on the cells in [3] is quite stringent.…”

“…More recently, problems of the kind (1) and (3) have been studied in [3], where it is shown that, under the a priori requirement that all the cells of the partitions are convex, the honeycomb conjecture holds true under the form (4) lim k→+∞ |Ω| 1/2 k 1/2 M k (Ω) = h(H) , lim k→+∞ |Ω| 1/2 k 3/2 m k (Ω) = h(H) , where H denotes the unit area regular hexagon. Clearly, the convexity assumption made on the cells in [3] is quite stringent. Nevertheless, as a first approach, it seemed reasonable to attack the problem under this restriction, since, also in the case of perimeter minimizing partitions, the case of convex polygonal cells was much simpler and indeed it was settled a long time before the celebrated result by Hales [10] (see Fejes Tóth [9]).…”

Optimal partitioning of an interval and applications to Sturm-Liouville eigenvalues

“…that can be builded along the road I, with problem (6) the builder is looking for the best position to place the facilities so as to minimize the (average or maximum) distance the people have to cover to reach the nearest facility. Theorem 2.11 then applies: for both problems the solution is unique and satisfies λ nj 1 (I j1 ) = λ nj 2 (I j2 ) for all 1 ≤ j 1 , j 2 ≤ n. Versions of these problems in higher dimension for the Laplace operator have applications to the phenomenon of the spatial segregation in reaction-diffusion systems [6,8,9,17]. By the way, many others spectral partition problems can be considered, such as the ones involving the eigenvalues of singular operators, nonlinear operators (i.e., the p-Laplacian for 1 < p < +∞), higher order operators (i.e., the bi-Laplacian) or fractional operators (i.e., the s-Laplacian for 0 < s < 1).…”

“…On one side, rigorous studies of partition problems require advanced mathematical tools; on the other, optimal partitions arise in several concrete applications, such as in discrete allocation problems, in statistical decision theory, and in phenomena of spatial segregation in reaction-diffusion systems. For some works on the subject one can consult [4,6,8,9,12,13,14,17,21] and the references therein.…”