Iterative Thresholding Methods for Longest Minimal Length Partitions

TL;DR

This paper presents two iterative methods to solve the longest minimal length partition problem, demonstrating that discs and balls are optimal solutions in 2D and 3D cases through numerical experiments.

Contribution

Introduction of novel iterative thresholding methods combining heat flow, auction dynamics, and threshold dynamics for partition problems.

Findings

Methods confirm discs and balls as optimal solutions in 2D and 3D.

Numerical experiments validate the effectiveness of the proposed algorithms.

Results support the conjecture regarding optimal shapes in the partition problem.

Abstract

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.