Facility location on manifolds Facility location studies optimal placement of a certain number $n$ of points (facilities) in some region $R$. (https://en.wikipedia.org/wiki/Facility_location_problem)
The minimax facility location problem finds $n$ locations in $R$ for the facility points such that: if $P$ is any point in $R$ and $d(P)$ is the distance from $P$ to the closest facility point, then the highest value of $d(P)$ as $P$ ranges over $R$ is minimized.
The dispersal problem places $n$ points in $R$ such that the minimum pair-wise distance among them is maximized.
Question: Is this claim true: "If the region $R$ is a closed and homogeneous manifold (such as a sphere or $S^1 \times S^1$) with a metric, then, for any $n$, if a given configuration of $n$ facility points solves one of the above problems, then, it automatically solves the other problem for that $n$"?
If the claim holds, does its converse also hold — "if both above problems have same answer for any $n$ in some $R$, then, $R$ is a closed and homogeneous manifold"?
Note: The earlier version of the post had a confused formulation of the maximin problem. Now, maximin has been replaced by dispersion.
 A: I believe the answer is No, the claim is not true.
At least not if my reformulation of the question is correct.
What the OP calls dispersion is usually viewed as optimal packing
of $n$ congruent circles, also known as the Tammes problem
(e.g., this MO question).
The OP's minimax facility location problem I believe can be viewed
as optimal cover by congruent disks, with the distance from any
point $p \in R$ to a facility at most the radius of a covering disk.
So the OP's question is essentially:

Is the best packing configuration also the best covering configuration?

In the plane this is true, a result of Richard Kershner from 1939.
So the OP's question is a natural extension.
Here is why I think that in more general situations, the claim is false.
Consider packing and covering the flat square torus by congruent circles/disks. There has been considerable work on the packing question,
a variant of the Tammes problem. There has been much less work on covering.
However, the two papers cited below indicate that the optimal configurations are not the same for $n=3$.
First the optimal packing. Note the asymmetry:
     
Next the optimal cover. Note the symmetry.
     
I could be misinterpreting these two papers, but if not, then
they provide an example showing the optimal packing and covering
configurations are, in general, not identical.

Brandt, Madeline, William Dickinson, AnnaVictoria Ellsworth, Jennifer Kenkel, and Hanson Smith. "Optimal packings of two to four equal circles on any flat torus." Discrete Mathematics 342, no. 12 (2019): 111597. DOI.


Joós, Antal. "On covering the square flat torus by congruent discs." Australasian J. Combinatorics 75, no. 1 (2019): 113-126.
PDF download.

