Longest minimal length partitions

TL;DR

This paper investigates the maximal perimeter of minimal partitions with prescribed areas under volume constraints, introducing a numerical algorithm to approximate and analyze such partitions, with a focus on the influence of domain perturbations.

Contribution

It presents a novel numerical maximization algorithm for minimal partitions, including a new method to identify capacity-constrained Voronoi diagrams and analyze their properties.

Findings

The ball maximizes the perimeter of minimal partitions under volume constraints.

The algorithm effectively approximates minimal partitions and their perturbations.

New techniques for computing gradients of Voronoi cell areas and perimeters are introduced.

Abstract

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.