The general case of the 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 PROOF (in the plane): 1 . For each possible direction s of a vector , we clearly have, for each region, a line in associated with the vector of direction s that bisects that region. 2. 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. 3. 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. example 4. 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. example 5. Since d is a continuous function, by the intermediate-value theorem it must achieve a value of 0 for some direction. |
