Covering points on a sphere with a disk

geometry general-topology geometric-topology tiling

Solution 1:

It turns out that the answer to my first question is really very simple.

Suppose you pick the center of the disk randomly from a uniform distribution on the sphere. Appealing to symmetry, we may infer that the probability that a given site lies within the disk is precisely the fraction of the surface area of the sphere covered by the disk; if the areas of the disk and the sphere are $a$ and $A$ respectively, this probability is $a/A$. By linearity of expectation, the expected number of sites contained in a randomly chosen disk is $ma/A$. Therefore, there must exist some disk which contains at least this many points.

The second question remains open, namely the problem of covering as many as possible of $m$ sites in a unit ball with a smaller ball of radius $r < 1$.

Edit: I hate to edit merely to bump this to the front page, but I wanted to use this solution to answer another question on math.SE, and I'd rather not do that if it has a negative score. For all I know, it might have an error that I haven't noticed. The one person who downvoted did not leave a reason; can anyone else let me know if this solution is incorrect?

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