The problem of finding the smallest number of copies of a certain shape that can be placed into a space to make fitting another copy impossible

Question

Packing problems often ask for the largest number of some identical shape that can fit in a given space without overlapping, if they are placed optimally.

I'm interested in the opposite question:

Q. Given a shape and a space, what is the smallest number of copies of this shape that can be placed in the space in such a way that any additional copy must necessarily overlap one of the previous ones?

Another, cuter, way to formulate this question is as follows:

Q. What is the smallest number of shapes you can always fit into the space, even if you are very bad at packing?

I've tried searching for "adversarial packing" without results, and most results of searches including "worst packing" are about finding shapes that have particularly bad packing densities.

Has anyone worked on problems like this, and what terms are useful in searching for their results?

The specific problem I'm primarily interested in is the number of discs of a certain size that are guaranteed to fit on a unit sphere. It might be that for this particular instance of the problem there is some relation between classical tight packings, coverings and the loose packings I ask about. After reading the answers to this question Optimal sphere packings ==> Thinnest ball coverings? about the relationship between sphere packings and ball coverings I'm cautious about being too optimistic any such relationship exists. Even if the problem formulated for discs on spheres can be translated to a classical packing problem or a covering problem, I'm still interested in the general case for other shapes and spaces.

I've also found Is there an "accepted" jamming limit for hard spheres placed in the unit cube by random sequential adsorption? , but this question assumes the shapes are being placed randomly, and not in a way that maximally reduces their number.

Answer 1

On the sphere and for disks of radius $r$, your problem is equivalent to a covering problem with disks of radius $2r$.

Given a collection of $k$ centerpoints of open $r$-disks on the sphere, you can fit one more open $r$-disk if and only if there is a point on the sphere that is at least $2r$ away from all $k$ centerpoints. Or equivalently, open $2r$-disks centered at those $k$ points do not cover the sphere.

You can probably generalize this to bounded areas, such as placing disks inside a rectangle, by adjusting the area boundary suitably. For shapes other than disks things might be (much) more complicated!