Fix two sets X and Y in the plane, each the interior of a curve, such that their closures don't intersect. How would one go about finding points x in X and y in Y such that the distance between X and Y is minimal?
Furthermore, say we have a bunch of sets X1, ..., Xn, each the interior of a curve, with nonintersecting closures. Let d(i,j) be the minimal distance between Xi and Xj. How can we find the minimal nonzero d(i,j), that is, the minimal distance between any two sets with nonintersecting closures? (In particular, there should be a faster algorithm than computing all the d(i,j). I suspect O(n log n) of the d(i,j) need to be computed although I have no idea why I'm saying this.)
The problem that inspired this is the following geographic one. What two states in the US that don't border each other are closest together? I think I know the answer; I'll post about that later.