  Hypervolume is widely used in the evolutionary multi-objective optimization
(EMO) field to evaluate the quality of a solution set. For a solution set with
$\mu$ solutions on a Pareto front, a larger hypervolume means a better solution
set. Investigating the distribution of the solution set with the largest
hypervolume is an important topic in EMO, which is the so-called hypervolume
optimal $\mu$-distribution. Theoretical results have shown that the $\mu$
solutions are uniformly distributed on a linear Pareto front in two dimensions.
However, the $\mu$ solutions are not always uniformly distributed on a
single-line Pareto front in three dimensions. They are only uniform when the
single-line Pareto front has one constant objective. In this paper, we further
investigate the hypervolume optimal $\mu$-distribution in three dimensions. We
consider the line- and plane-based Pareto fronts. For the line-based Pareto
fronts, we extend the single-line Pareto front to two-line and three-line
Pareto fronts, where each line has one constant objective. For the plane-based
Pareto fronts, the linear triangular and inverted triangular Pareto fronts are
considered. First, we show that the $\mu$ solutions are not always uniformly
distributed on the line-based Pareto fronts. The uniformity depends on how the
lines are combined. Then, we show that a uniform solution set on the
plane-based Pareto front is not always optimal for hypervolume maximization. It
is locally optimal with respect to a $(\mu+1)$ selection scheme. Our results
can help researchers in the community to better understand and utilize the
hypervolume indicator.
