In this paper, we introduce two iterative methods for longest minimal length partition problem, which asks whether the disc (ball) is the set maximizing the total perimeter of the shortest partition that divides the total region into sub-regions with given volume proportions, under a volume constraint. The objective functional is approximated by a short-time heat flow using indicator functions of regions and Gaussian convolution. The problem is then represented as a constrained max-min optimization problem. Auction dynamics is used to find the shortest partition in a fixed region, and threshold dynamics is used to update the region. Numerical experiments in two-dimensional and three-dimensional cases are shown with different numbers of partitions, unequal volume proportions, and different initial shapes. The results of both methods are consistent with the conjecture that the disc in two dimensions and the ball in three dimensions are the solution of the longest minimal length partition problem.

翻译：本文针对最长最小长度分割问题提出了两种迭代求解方法。该问题探究在体积约束下，圆盘（或球体）是否为使分割总周长最大化的集合，其中分割需将整个区域划分为具有给定体积比例的子区域。目标泛函通过区域指示函数与高斯卷积的短时热流进行逼近。该问题随后被表述为一个带约束的最大-最小优化问题。其中，拍卖动力学用于在固定区域内寻找最短分割，而阈值动力学则用于更新区域。文中展示了二维与三维情形下的数值实验，涵盖了不同分割数量、不等体积比例及不同初始形状的情况。两种方法所得结果均与以下猜想一致：二维圆盘与三维球体是最长最小长度分割问题的解。