# On the honeycomb conjecture for a class of minimal convex partitions

## TL;DR

This paper proves that the hexagonal honeycomb is asymptotically optimal for a broad class of convex partition problems minimizing various shape functionals, extending the honeycomb conjecture to new contexts.

## Contribution

It establishes the honeycomb conjecture for convex partitions under general shape functionals satisfying specific assumptions, including Cheeger constant and logarithmic capacity.

## Key findings

- Hexagonal honeycomb is asymptotically optimal for convex partitions.
- Results apply to Cheeger constant and logarithmic capacity.
- Conditions identified for extending to the first Dirichlet eigenvalue.

## 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 $F$ 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 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.