最长最小长度划分
Longest minimal length partitions
摘要：
本文提供了数值证据表明，在体积约束下的球是集最大化的最小周长划分成具有规定的区域的细胞的周长。我们引入了一个数值最大化算法，它在每次迭代中执行多个优化步骤来近似最小分区。使用这些分区，我们计算扰动的域，增加了最小周长。最优划分算法的初始化使用容量约束的Voronoi图。提出了一种新的算法来识别这样的图，通过计算Voronoi单元的面积和周长相对于Voronoi点的梯度。
英文摘要：
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.
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