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Well hello there,
This thread unearthed an old math memory...
The Ham Sandwich Theorem:
Regardless of the distribution of ham and cheese in a sandwich, you can use one slice to divide the sandwich into two parts containing equal parts of bread, ham and cheese.
The general case of the Ham-Sandwich Theorem says that if we have n regions in n-dimensional space, then there is some hyperplane, which cuts each exactly in half, measured by volume.
The general proof is suggested by the argument in the two-dimensional case. There, for each possible direction s for the cut, we clearly have, for each region, a line in direction s bisecting that region. But the two lines for the two regions are offset by some distance d(s). We'd like to find a direction with d(s) = 0.
Note that, if we have rotate our direction by 180 degrees, we get back to the same pair of bisecting lines, but they now have the opposite orientation. Adopting the convention that the distance d between the lines is a signed quantity depending on the orientation, we see that d(s+180) = -d(s). Thus it is clear that d is neither always positive nor always negative. Since d is a continuous function, by the intermediate-value theorem it must achieve a value of 0 for some direction.
Thinking of the circle of directions as the unit circle in the plane, we might write -s instead of s+180 for the opposite direction. Thus we have a function d from the circle of directions to the real numbers, with the property that antipodal points map to negatives of each other: d(-s) = -d(s). Such a function must be zero somewhere.
*looks around*
Anyone still awake?
-GH
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