For any point on the globe, I believe there is (by the mean value theorem) at least one great circle containing that point and dividing the world's land area (or water mass, or population, whatever) into equal halves.
Assuming that the stuff to be bisected has no nontrivial symmetries, can we say anything interesting about the set of such circles? For example, do their poles all lie on a finite number of circles?
(As usual, I hope for help with the tags)