[EM] Demonstrations of CW SU claims

[MIKE OSSIPOFF]

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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