  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.
