  This Thesis aims to highlight some isoperimetric questions involving the,
so-called, $N$-clusters. We first briefly recall the theoretical framework we
are adopting. This is done in Chapter one. In chapter two we focus on the
standard isoperimetric problem for planar $N$-cluster for large values of $N$
and we provide an equidistribution energy-type results under some suitable
assumption. The third Chapter is devoted to a stability results of the
hexagonal honeycomb tiling. Finally in the fourth Chapter we consider a
generalization of the Cheeger constant, defined as a minimization of a suitable
energy among the class of the $N$-clusters. We show how this problem is related
to the optimal partition problem for the first Dirichlet eigenvalue of the
Laplacian introduced by Caffarelli and Fang-Hua Lin in 2007. We conclude, in
Chapter five, with some remarks and some possible future direction of
investigation.
