Let r > 0. Suppose a closed set F in a k-sphere does not lie in any ball (in the sphere) of radius r (in great circle distance). What's the minimum diameter F can have? Assume the sphere is a unit sphere and, if it helps, r = pi/4. I've computed a lower bound on the diameter but I bet others have studied this problem more thoroughly than I and may have computed a tighter lower bound than mine.