Why does the CW always maximize SU, with city block distance?
This is completely straightforward. Say we start at the median point, the point where the equal sections intersect. (The equal sections are the sections of the issue space that divide the voters into 2 parts).
Say we start moving away from that point along one of the co-ordinate axes, the x axis. That motion in that direction obviously increases the city block distance to every voter on the other side of the equal section that we're moving away from. Then, when we've passed a few voters on our way out, reached an x co-ordinate greater than theirs, we're then increasing our city block distance to them too.
So as soon as we begin moving out in the x direction, we're increasing our distance to more voters than we're decreasing our distance to. We're decreasing out distance to every voter whose x co-ordinate is greater than ours.
And, with city block distance, all these rates of distance increase & decrease are the same.
Now, say we stop and begin moving in the positive y direction. Again, we're starting at the y equal section and moving away from it, and it's the same as I described above, when we were movingg away from the x equal section. Likewise if we later stop and begin moving away from the z equal section.
Of course if the voters aren't densely packed, we could move away from the x equal section for some small distance before we pass a voter's x co-ordinate. Over that distance our SU isn't changing.
But neither have we made it so more voters are closer to the candidate at the median point than to us.
Why does weak radial symmetry guarantee that the CW maximizes Su, even with Euclidean distance?
Say we divide the issue space into thin spherical shells, concentric about the center of the radially symmetrical distributiion. Consider one of those shells.
Consoder one pair identical rays in opposite directions. Or, rather, a pair of very thin conical bundles of rays in opposite directions, with every ray having a population density distribution the same as its opposite.
Where each of those 2 cones intersects the spherical shell, the population in the intersection is the same. Those cone-sphere intersections will be referred to as the intersections.
Say we start at the distribution center and begin moving away from it along a straight line, which I'll call the departure line.
As soon as we've moved any distance along the departure line, the angle between the departure line and the line from us to the intersection we're moving away from is less than the angle between the departure line and the line from us to the intersection we're moving toward.
That means that the cosine of the angle between the departure line and the line from us to the intersection we're moving away from is greater than the cosine of the angle between the departure line and the intersection we're moving toward.
And that means that we're moving away from the intersection that we're moving away from at a higher rate than the rate at which we're moving toward the intersection that we're moving toward.
That's true of the opposite intersections on the spherical shell, of any 2 very thin conical bundles of rays in opposite directions fromt he center of the distribution, and for any departure line.
And it's also true for any spherical shell about the distribution center.
That means that when we move along a line away from the distribution center, we're increasing our summed distance from population, since it's true of every pair of thin cones leading from the center in any particular shell, and since it's true for every shell.
That assumes that it's a shell that we're inside of. If we're outside the shell and moving away from the center, then it's obvious that we're increasing our summed distance from that shell's population too.
So our SU can only decrease if we move away from the distribution center.
Sometimes I've referred to the median point. There's always a median point. The equal sections will always intersect at a point. That's true, for example, if they're 2 perpendicular lines, or 3 perpendicular planes.
Is there always an equal section for each dimension? What if there's an odd number of voters, or 2 voters whose x co-ordinate is the same? For that to affect the outcome of an election the race would have to be improbably close. For practical purposes it's safe to say that there's an equal sectiion for each dimension.
Mike Ossipoff
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