Gerrymandering and equal-population districts

On constraining gerrymanderers with convexity requirements (pdf):

a gerrymanderer can always create equal sized convex constituencies that translate a margin of k voters into a margin of at least k constituency wins. Thus even with a small margin a majority party can win all constituencies. Moreover there always exists some population distribution such that all divisions into equal sized convex constituencies translate a margin of k voters into a margin of exactly k constituencies. Thus a convexity constraint can sometimes prevent a gerrymanderer from generating any wins for a minority party.

The current congressional districts in Iowa are a bit wrapped around each other; an initial districting proposal with more "compact" districts was replaced with this one, which had more nearly equal numbers of voters in each district as of the 2000 census. (The numbers in the initial plan were themselves so close that there's simply no way that 10 years of population movements wouldn't expand the variance by a large factor.) As long as we have single-member districts, and political minorities are going to be stuck with a single representative chosen by others in their district, it seems proper to me to favor a bit of homogeneity in each district, and "compactness" may function as a proxy for that. (The "population distribution such that all divisions into equal sized convex constituencies translate a margin of k voters into a margin of exactly k constituencies" is a theoretical curiosity, and is not likely in the world of geographical homophily in which we actually live.)

The absolute equality of district size is something of a misguided fetish. If you drew congressional districts largely at random with only a vague interest in keeping populations within about 50% of each other, I expect that congressional elections would play out similarly to districts that were more punctiliously equalized; if the former were able to be drawn with more homogeneity than the latter, they would leave most people better represented by "their" representative. In actual practice, of course, you would have Democrats drawing more populous Republican districts and vice versa; I think the best argument for keeping Congressional districts approximately the same size is that it places a constraint on gerrymandering. In addition to the homogeneity motive, "compactness" has the virtue of creating an — in some sense random — additional constraint on people who are likely, left to their own devices, to be worse than random. While this paper shows that convexity and equal populations aren't themselves sufficient constraints, I'm still tempted by the intuition that something like convexity, combined with other constraints — probably related to other political lines — would have a salutary effect on protecting us from a self-propagating political class. (That intuition wouldn't have expected the results of this paper, though. If I were precise in my statement, I could well be proved wrong.)