A previous post mentioned the Borsuk–Ulam theorem. A corollary (for the case of a circle) is the pancake theorem: given two plane regions, there is a cut (line) which divides both in half by area.
The example shows a cut dividing both South America and Africa in half (the theorem doesn't tell us how to find the cut; I used simulated annealing).
The corollary for the 2-sphere is the ham sandwich theorem: given three 3-dimensional objects, there is a cut (plane) which divides all three in half by volume.
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