It's occurred to me that two conditions might be enough to attain some kind of fair division: 1) Each region should be convex (or spherically convex on earth); and 2) Each region should be sufficiently circular. 2) refers to the fact that any simply-connected bounded region has a well-defined "eccentricity" — the eccentricity of the unique best-approximating filled ellipse. Requiring that the eccentricity lie below some maximum value makes the region "sufficiently circular". —Dan -----Original Message-----

From: "Keith F. Lynch" <kfl@KeithLynch.net> Sent: Apr 5, 2018 7:52 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Gerrymandering

It's been nearly four years since we discussed gerrymandering.

I've been thinking about it lately, since it's been back in the news, and since I've come up with what I think is an original way to measure it.

Last time we discussed it, the discussion was all about the borders, e.g. minimizing the variation in their curvature, or minimizing the ratio between the square of the perimeter and the area enclosed.

But that's obviously not the right measure, since making a border frizzy on a centimeter scale would greatly change such a measure without changing the real amount of gerrymandering at all.

My proposal is to start with a database listing, to high precision (e.g. 10 meters) the home location of everyone in the state. The gerrymandering in a district would be measured by summing the squares of the distances between each two people in the district. The gerrymandering in a state would be measured by summing the squares of the gerrymandering numbers for each district in the state. (Of course the number of districts is kept constant, and their populations are kept as equal as possible, i.e. the most populous district can have at most one more person than the least populous district.)

There's probably no practical algorithm for finding the unique lowest score. But everyone would be free to draw boundaries however they liked, so long the populations were equal, and submit their scores. Whoever came up with the lowest score before the deadline would have their submission accepted.

Simulated annealing would probably be a good way to find low-score redistricting plans.

This is fairly CPU-intensive, but I think today even laptop computers could easily handle it. With N people in a district, there would be N*(N-1)/2 comparisons to make.

Is census data available that gives everyone's home location to within ten meters? Of course all other information (name, age, income, citizenship, etc.) would be stripped for privacy reasons.

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