  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.
