The ordinary circle packing problem in the variant with equal radii asks for the largest radius $r_{max}$ that allows placing $n$ non-overlapping circles with radius $r_{max}$ e.g. in the unit square, in the unit circle or, on the unit sphere (Tammes' problem).

Now, I would like to solve a somehow opposite problem:

Question:given a number $n\in\mathbb{N}$, what is the smallest radius $r_{min}\in\mathbb{R}^+$ that permits a non-overlapping, rigid placement of $n$ circles with radius $r_{min}$ in the unit square, or in the unit circle or, on the unit sphere?

Under a rigid configuration I understand a configuration, where every open halfplane, resp. hemisphere defined by a hyperplane through a circle's center contains at least one contact point with another circle or, with the boundary of the containing region.

Are there already algorithms and/or theoretical results available for that problem?