We continue from Bounds for the Dispersal Problem in convex regions and Bounds for minimax facility location in a convex region
Let us consider the classes of problems:
- Given a convex region $R$ and an integer $n$, to pack $n$ discs into $R$ such that the smallest of the discs has the maximum possible radius.
- Given $R$ and $n$, to cover $R$ with $n$ disks such that the radius of the largest disk used is minimized.
Definition: In what follows, we refer to packing (covering) layouts of disks as per above specifications as optimal packing (covering) layouts.
Question: It is obvious from https://erich-friedman.github.io/packing/cirincir/ that the optimal packing layout when $R$ is a disk need not always have the radii of all the disks in the packing layout the same (see $n$ = 8, 9, 13, 20...). However, in view of https://erich-friedman.github.io/packing/circovcir/
Is it that when $R$ is a disk being covered with any number $n$ of disks, all disks in an optimal covering layout will automatically be of same radius?
Remarks: if $R$ is a general convex region (not a disk) for which we try to find the optimal covering layout with $n$ disks, then, all disks need not be of same radius in what seems to be a limited number of cases ( eg: when $R$ is a triangle and n = 2 or 5 as given in https://erich-friedman.github.io/packing/cirincir/ ). However, it seems likely that for large enough n, for any convex $R$, all disks in an optimal covering layout do have the same radius. This appears to mark a difference with packing where it appears possible to have differences in radius among disks in an optimal packing layout for even some large values of $n$ even when $R$ is itself a disk (eg: $n$ = 8, 9, 13, 20 in https://erich-friedman.github.io/packing/cirincir/).
Yet another point of possible difference is that for some $R$, in an optimal packing layout, for any $n$, all disks used could be of different radius - as appears to happen when $R$ is a long and thin isosceles triangle - but such large numbers of radius values of disks don't seem to happen in optimal covering layouts.
Further question: In 3 and higher dimensions, when any convex region $R$ is optimally covered with $n$ solid spheres, will all spheres necessarily be of same radius? The guess is that the difference between packing and covering will be more in higher dimensions.
