For city block distance and Euclidean distance
Say the origin of the co-ordinate system is the voter-median point.
Yesterday, though I said everything that was needed, I didn't say it as neatly as I could. I'm in a hurry again this time, but I'll try to be a little clearer about why the voter-median candidate maximizes SU with city-block distance.
If you're at a point with a positive X co-ordinate, and you move in the positive X direction, then you're obviously increasing your city-block distance from all the candidates whose X co-ordinate is less than yours at exactly the same rate as you're decreasing your distance to the candidates whose X co-ordinate is greater than yours.
If your X co-ordinate is greater than that of the voter-median point, and you're moving in the positive X direction, then you're obviously moving away from more candidates than you're moving toward--at the same rate. That means that you're increasing your summed distance to thet candidates.
That's the same for every issue-dimension, and for every point away from the voter-median point.
Euclidean distance:
Say that, for all straight lines passing through some central point P, the distribution of voter population density is the same in both directions from P.
Consider a spherical shell with P as its center. You're moving away from P along some line. Consider equal-size rings drawn on the inner surface of the sphere about the points where that line intersects the sphere.
When you've gone some distance away from P, continuing in the same direction away from P, the cosine of the angle between the line on which you're moving and the ring toward which you're moving greater than the cosine of the angle between the line on which you're moving and the ring away from which you're moving. That's because though both rings have the same radius, but you're closer to the one toward which you're moving.
That means that you're moving toward one ring at a lower rate than you're moving away from the other. For those rings, you're increasing your summed distance to candidates on the rings. That's true of other rings on the sphere about those points, and it's true for other spherical shells about P.
Mike Ossipoff
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