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@DeaneYang Since evolution by the heat equation is linear and commutes with rotations, spherical harmonics are eigenfunctions. I.e. $\phi[\sum_l f_l(x)] = \sum \phi_l f_l(x)$. If the initial measure has center of mass at zero, then $f_1 = 0$ and so the evolved measure would also have center of mass at zero.
Sorry I misread your question the first time. So it sounds like you're asking about the diameter of a random point set? I unfortunately don't recall anything helpful to point you to for that question. A quick search on Google Scholar turned up some papers looking at the diameter of a set of uniformly random point drawn from the ball and from a compact plane set. Maybe those could be helpful. (doc.rero.ch/record/311708/files/10687_2007_Article_38.pdf, doi.org/10.1239/aap/1019160946)
And arXiv's traceback helpfully reminded me I have previously linked to this paper on MO on a different question. So that question might be relevant too: mathoverflow.net/questions/245027/…
I think this paper might be of help: Random Point Sets on the Sphere --- Hole Radii, Covering, and Separation (arxiv.org/abs/1512.07470). Specifically, you're asking about the "separation" part. I think the paper mainly deals with the asymptotics for large n, but might have some info for finite n.
@AntonPetrunin Good question (I see now it was also raise by disznoperzselo in the comments to the question). It would be nice to have a solution where Eve's best guess is correct with a lower probability, but I don't see how to adapt this solution in that direction. I don't see any reason that the probability couldn't be improved, but maybe the improved protocol will not be as nice and symmetric.
This question, mathoverflow.net/questions/78572/…, cites a result by Berger which I cannot access, but I read it is saying that all such ngon billiard paths have equal maximal perimeter.
Your argument about dislocations reminded me of the theory that Bowick, Nelson, and Travesset (arxiv.org/abs/cond-mat/9911379 as well as follow ups) described for using elasticity theory to give a continuum limit description of the way dislocations and disclinations interact in a 2D crystal on a curved surface.
Have you tried calculating the density for the obvious double-lattice candidate for circular segments? I'm pretty sure it should easily beat the the circle-packing density. Even the obvious lattice candidate probably beats it, no?
Another counter-example is sudoku completion. The finite version is NP-complete. An infinite ($\omega^2\times\omega^2$) sudoku board with finitely many squares filled in can be trimmed to a finite instance containing all the filled in squares, completed, and then extended.
