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 optimization 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.
Highlights
In [18] the authors answer a question raised by Polya in [38] and prove that among planar convex sets of given area the disk maximizes the length of the shortest area-bisecting curve
SF (Ω, c) is one subset ω ⊂ Ω which minimizes the relative perimeter PerΩ(ω) when the measure |ω| is fixed to c|Ω|
In this work we propose a new way of computing capacity-constrained Voronoi diagrams by explicitly computing the gradients of the areas of the Voronoi cells with respect to variations in the Voronoi points
Summary
In [18] the authors answer a question raised by Polya in [38] and prove that among planar convex sets of given area the disk maximizes the length of the shortest area-bisecting curve. In other words P I(Ω, c) is the minimal total relative perimeter of a partition with volume constraints given by c. Compute reliably a numerical approximation of the shortest partition SP (Ω, c) once the domain Ω and the constraints vector c are given. We underline that the maximization algorithm approximates solutions to a max-min problem, and the optimal partitioning algorithm presented in Section 3.1 is run at every iteration.
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