An earlier question: Facility location on manifolds
A possibly related earlier post: Cutting convex regions into equal diameter and equal least width pieces - 2
The minimax facility location problem finds $n$ locations in some domain $R$ 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.
Let us assign each point on $R$ to that facility point closest to it, Define $D(i)$ as the maximum distance to the $i$-th facility from among points on $R$ assigned to that facility.
Question: If $R$ is a convex region of unit diameter and $n$ facility points are to be located within it, what is the upper bound on the variation in $D(i)$ (say, the ratio of the largest value to the smallest) in a solution of the minimax problem? Which shape of $R$ achieves this bound? What is the dependency of this variation on $n$ and dimensionality of $R$?
I know of no example where $D(i)$ shows variation over $i$ in an optimal solution to the minimax location problem.
Note: An analogous question can also be asked with a road network replacing the convex region, distances between points on the network being path lengths and each network point assigned to the closest facility point. The p-median problem might be related.
