  This article provides numerical evidence that under volume constraint the
ball is the set which maximizes the perimeter of the least-perimeter partition
into cells with prescribed areas. We introduce a numerical maximization
algorithm which performs multiple optimizations steps at each iteration to
approximate minimal partitions. Using these partitions we compute perturbations
of the domain which increase the minimal perimeter. The initialization of the
optimal partitioning algorithm uses capacity-constrained Voronoi diagrams. A
new algorithm is proposed to identify such diagrams, by computing the gradients
of areas and perimeters for the Voronoi cells with respect to the Voronoi
points.
